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Meanwhile, construction continued on the Tobin Bridge approach. By the time all parties agreed on the I-93 design, construction of the Tobin connector, today known as the "City Square Tunnel" for a Charlestown area it bypasses, was far along, significantly adding to the cost of constructing the US Route 1 interchange and retrofitting the tunnel.
Boston blue clay and other soils extracted from the path of the tunnel were used to cap many local landfills, fill in the Granite Rail Quarry in Quincy, and restore the surface of Spectacle Island in the Boston Harbor Islands National Recreation Area.
The Storrow Drive Connector, a companion bridge to the Zakim, began carrying traffic from I-93 to Storrow Drive in 1999. The project had been under consideration for years, but was opposed by the wealthy residents of the Beacon Hill neighborhood. It was finally accepted because it would funnel traffic bound for Storrow Drive and downtown Boston away from the mainline roadway. The Connector ultimately used a pair of ramps that had been constructed for Interstate 695, enabling the mainline I-93 to carry more traffic that would have used I-695 under the original Master Plan.
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When construction began, the project cost, including the Charles River crossing, was estimated at $5.8 billion. Eventual cost overruns were so high that the chairman of the Massachusetts Turnpike Authority, James Kerasiotes, was fired in 2000. His replacement had to commit to an $8.55 billion cap on federal contributions. The total expenses eventually passed $15 billion. Interest brought this cost to $21.93 billion.
Engineering methods and details.
Several unusual engineering challenges arose during the project, requiring unusual solutions and methods to address them. At the beginning of the project, engineers had to figure out the safest way to build the tunnel without endangering the existing elevated highway above. Eventually, they created horizontal braces as wide as the tunnel, then cut away the elevated highway's struts, and lowered it onto the new braces.
Three alternative construction methods were studied with their corresponding structural design to address existing conditions, safety measures, and constructability. In addition to codified loads, construction loads were computed to support final design and field execution.
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Final phases.
On January 18, 2003, the opening ceremony was held for the I-90 Connector Tunnel, extending the Massachusetts Turnpike (Interstate 90) east into the Ted Williams Tunnel, and onwards to Boston Logan International Airport. The Ted Williams tunnel had been completed and was in limited use for commercial traffic and high-occupancy vehicles since late 1995. The westbound lanes opened on the afternoon of January 18 and the eastbound lanes on January 19.
The next phase, moving the elevated Interstate 93 underground, was completed in two stages: northbound lanes opened on March 29, 2003, and southbound lanes (in a temporary configuration) on December 20, 2003. A tunnel underneath Leverett Circle connecting eastbound Storrow Drive to I-93 North and the Tobin Bridge opened December 19, 2004, easing congestion at the circle. All southbound lanes of I-93 opened to traffic on March 5, 2005, including the left lane of the Zakim Bridge, and all of the refurbished Dewey Square Tunnel.
By the end of December 2004, 95% of the Big Dig was completed. Major construction remained on the surface, including construction of final ramp configurations in the North End and in the South Bay interchange, and reconstruction of the surface streets.
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The final ramp downtown—exit 16A (formerly 20B) from I-93 south to Albany Street—opened January 13, 2006.
In 2006, the two Interstate 93 tunnels were dedicated as the Thomas P. O'Neill Jr. Tunnel, after the former Democratic speaker of the House of Representatives from Massachusetts who pushed to have the Big Dig funded by the federal government.
Coordinated projects.
The Commonwealth of Massachusetts was required under the Federal Clean Air Act to mitigate air pollution generated by the highway improvements. Secretary of Transportation Fred Salvucci signed an agreement with the Conservation Law Foundation in 1990 enumerating 14 specific projects the state agreed to build. This list was affirmed in a 1992 lawsuit settlement.
Projects which have been completed include:
However, some projects were removed:
Surface treatments.
Some surface treatments that were part of the original project plan were dropped due to the massive cost overruns on the highway portion of the project.
$99.1 million was allocated for mitigating improvements to the Charles River Basin, including the construction of North Point Park in Cambridge and Paul Revere Park in Charlestown. The North Bank Bridge, providing pedestrian and bicycle connectivity between the parks, was not funded until the American Recovery and Reinvestment Act of 2009. Nashua Street Park on the Boston side was completed in 2003, by McCourt Construction with $7.9 million in funding from MassDOT.
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As of 2017, $30.5 million had been transferred to the Massachusetts Department of Conservation and Recreation to complete five projects. Another incomplete but required project is the South Bank Bridge over the MBTA Commuter Rail tracks at North Station (connecting Nashua Street Park to the proposed South Bank Park, which is currently a parking lot under the Zakim Bridge at the Charles River locks).
Improvements in the lower Charles River Basin include the new walkway at Lovejoy Wharf (constructed by the developer of 160 North Washington Street, the new headquarters of Converse), the Lynch Family Skate Park (constructed in 2015 by the Charles River Conservancy), rehabilitation of historic operations buildings for the Charles River Dam and lock, a maintenance facility, and a planned pedestrian walkway across the Charles River next to the MBTA Commuter Rail drawbridge at North Station (connecting Nashua Street Park and North Point Park). MassDOT is funding the South Bank Park, and replacement of the North Washington Street Bridge (construction Aug 2018–23).
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EF Education is funding public greenspace improvements as part of its three-phase expansion at North Point. Remaining funding may be used to construct the North Point Inlet pedestrian bridge, and a pedestrian walkway over Leverett Circle. Before being replaced with surface access during the reconstruction of the Science Park MBTA Green Line station, Leverett Circle had pedestrian bridges with stairs that provided elevated access between the station, the Charles River Parks, and the sidewalk to the Boston Museum of Science. The replacement ramps would comply with Americans with Disabilities Act requirements and allow easy travel by wheelchair or bicycle over the busy intersection.
Public art.
While not a legally mandated requirement, public art was part of the urban design planning process, and later design development work, through the Artery Arts Program. The intent of the program was to integrate public art into highway infrastructure (retaining walls, fences, and lighting) and the essential elements of the pedestrian environment (walkways, park landscape elements, and bridges). As overall project costs increased, the Artery Arts Program was seen as a potential liability, even though there was support and interest from the public and professional arts organizations in the area.
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At the beginning of the highway design process, a temporary arts program was initiated, and over 50 proposals were selected. Development began on only a few projects before funding for the program was cut. Permanent public art that was funded includes: super graphic text and facades of former West End houses cast into the concrete elevated highway abutment support walls near North Station by artist Sheila Levrant de Bretteville; Harbor Fog, a sensor-activated mist, light and sound sculptural environment by artist Ross Miller in parcel 17; a historical sculpture celebrating the 18th and 19th century shipbuilding industry and a bust of shipbuilder Donald McKay in East Boston; blue interior lighting of the Zakim Bridge; and the Miller's River Littoral Way walkway and lighting under the loop ramps north of the Charles River.
Extensive landscape planting, as well as a maintenance program to support the plantings, was requested by many community members during public meetings.
Impact on traffic.
The Big Dig separated the co-mingled traffic from the Massachusetts Turnpike and the Sumner and Callahan tunnels. While only one net lane in each direction was added to the north–south I-93, several new east–west lanes became available. East–west traffic on the Massachusetts Turnpike/I-90 now proceeds directly through the Ted Williams Tunnel to Logan Airport and Route 1A beyond. Traffic between Storrow Drive and the Callahan and Sumner Tunnels still uses a short portion of I-93, but additional lanes and direct connections are provided for this traffic.
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The result was a 62% reduction in vehicle hours of travel on I-93, the airport tunnels, and the connection from Storrow Drive, from an average 38,200 hours per day before construction (1994–1995) to 14,800 hours per day in 2004–2005, after the project was largely complete. The savings for travelers was estimated at $166 million annually in the same 2004–2005 time frame. Travel times on the Central Artery northbound during the afternoon peak hour were reduced 85.6%.
A 2008 "Boston Globe" report asserted that waiting time for the majority of trips actually increased as a result of demand induced by the increased road capacity. Because more drivers were opting to use the new roads, traffic bottlenecks were only pushed outward from the city, not reduced or eliminated (although some trips are now faster). The report states, "Ultimately, many motorists going to and from the suburbs at peak rush hours are spending more time stuck in traffic, not less." The "Globe" also asserted that their analysis provides a fuller picture of the traffic situation than a state-commissioned study done two years earlier, in which the Big Dig was credited with helping to save at least $167 million a year by increasing economic productivity and decreasing motor vehicle operating costs. That study did not look at highways outside the Big Dig construction area and did not take into account new congestion elsewhere.
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Impact on property values.
Towards the end of the Big Dig in 2003, it was estimated that the demolition of the Central Artery highway would cause a $732 million increase in property value in Boston's financial district, with the replacement parks providing an additional $252 million in value. As a result of the Big Dig, a large amount of waterfront space was opened up, which is now a high-rent residential and commercial area called the Seaport District. The development of Seaport alone was estimated to create $7 billion in private investment and 43,000 jobs.
Operations Control Center (OCC).
As part of the project, an elaborate Operations Control Center (OCC) control room was constructed in South Boston. Staffed on a "24/7/365" basis, this center monitors and reports on traffic congestion, and responds to emergencies. Continuous video surveillance is provided by hundreds of cameras, and thousands of sensors monitor traffic speed and density, air quality, water levels, temperatures, equipment status, and other conditions inside the tunnel. The OCC can activate emergency ventilation fans, change electronic display signs, and dispatch service crews when necessary.
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Problems.
Leaks.
As far back as 2001, Turnpike Authority officials and contractors knew of thousands of leaks in ceiling and wall fissures, extensive water damage to steel supports and fireproofing systems, and overloaded drainage systems. Many of the leaks were a result of Modern Continental and other subcontractors failing to remove gravel and other debris before pouring concrete. This information was not made public until engineers at MIT (volunteer students and professors) performed several experiments and found serious problems with the tunnel.
On September 15, 2004, a major leak in the Interstate 93 north tunnel forced the closure of the tunnel while repairs were conducted. This also forced the Turnpike Authority to release information regarding its non-disclosure of prior leaks. A follow-up reported on "extensive" leaks that were more severe than state authorities had previously acknowledged. The report went on to state that the tunnel system had more than 400 leaks. A "Boston Globe" report countered that by stating there were nearly 700 leaks in a single section of tunnel beneath South Station. Turnpike officials also stated that the number of leaks being investigated was down from 1,000 to 500.
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The problem of leaks is further aggravated by the fact that many of them involve corrosive salt water. This is caused by the proximity of Boston Harbor and the Atlantic Ocean, causing a mix of salt and fresh water leaks in the tunnel. The situation is made worse by road salt spread in the tunnel to melt ice during freezing weather, or brought in by vehicles passing through. Salt water and salt spray are well-known issues that must be dealt with in any marine environment. It has been reported that "hundreds of thousands of gallons of salt water are pumped out monthly" in the Big Dig, and a map has been prepared showing "hot spots" where water leakage is especially serious. Salt-accelerated corrosion has caused ceiling light fixtures to fail (see below), but can also cause rapid deterioration of embedded rebar and other structural steel reinforcements holding the tunnel walls and ceiling in place.
Substandard materials.
Massachusetts State Police searched the offices of Aggregate Industries, the largest concrete supplier for the underground portions of the project, in June 2005. They seized evidence that Aggregate delivered concrete that did not meet contract specifications. In March 2006 Massachusetts Attorney General Tom Reilly announced plans to sue project contractors and others because of poor work on the project. Over 200 complaints were filed by the state of Massachusetts as a result of leaks, cost overruns, quality concerns, and safety violations. In total, the state has sought approximately $100 million from the contractors ($1 for every $141 spent).
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In May 2006, six employees of the company were arrested and charged with conspiracy to defraud the United States. The employees were accused of reusing old concrete and double-billing loads. In July 2007, Aggregate Industries settled the case with an agreement to pay $50 million. $42 million of the settlement went to civil cases and $8 million was paid in criminal fines. The company will provide $75 million in insurance for maintenance as well as pay $500,000 toward routine checks on areas suspected to contain substandard concrete.
In July 2009, two of the accused, Gerard McNally and Keith Thomas, both managers, pled guilty to charges of conspiracy, mail fraud, and filing false reports. The following month, the remaining four, Robert Prosperi, Mark Blais, Gregory Stevenson, and John Farrar, were found guilty on conspiracy and fraud charges. The four were sentenced to probation and home confinement and Blais and Farrar were additionally sentenced to community service.
Fatal ceiling collapse.
A fatal accident raised safety questions and closed part of the project for most of the summer of 2006. On July 10, 2006, concrete ceiling panels and debris weighing and measuring fell on a car traveling on the two-lane ramp connecting northbound I-93 to eastbound I-90 in South Boston, killing Milena Del Valle, who was a passenger, and injuring her husband, Angel Del Valle, who was driving.
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Immediately following the fatal ceiling collapse, Governor Mitt Romney ordered a "stem-to-stern" safety audit conducted by the engineering firm of Wiss, Janney, Elstner Associates, Inc. to look for additional areas of risk. Said Romney: "We simply cannot live in a setting where a project of this scale has the potential of threatening human life, as has already been seen".
The collapse and closure of the tunnel greatly snarled traffic in the city. The resulting traffic jams are cited as contributing to the death of another person, a heart attack victim who died en route to Boston Medical Center when his ambulance was caught in one such traffic jam two weeks after the collapse. On September 1, 2006, one eastbound lane of the connector tunnel was re-opened to traffic.
Following extensive inspections and repairs, Interstate 90 east- and westbound lanes reopened in early January 2007. The final piece of the road network, a high occupancy vehicle lane connecting Interstate 93 north to the Ted Williams Tunnel, reopened on June 1, 2007.
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On July 10, 2007, after a lengthy investigation, the National Transportation Safety Board found that epoxy glue used to hold the roof in place during construction was not appropriate for long-term bonding. This was determined to be the cause of the roof collapse. The Power-Fast Epoxy Adhesive used in the installation was designed for short-term loading, such as wind or earthquake loads, not long-term loading, such as the weight of a panel.
Powers Fasteners, the makers of the adhesive, revised their product specifications on May 15, 2007, to increase the safety factor from 4 to 10 for all of their epoxy products intended for use in overhead applications. The safety factor on Power-Fast Epoxy was increased from 4 to 16. On December 24, 2007, the Del Valle family announced they had reached a settlement with Powers Fasteners that would pay the family $6 million. In December 2008, Powers Fasteners agreed to pay $16 million to the state to settle manslaughter charges.
"Ginsu guardrails".
Public safety workers have called the walkway safety handrails in the Big Dig tunnels "ginsu guardrails", because the squared-off edges of the support posts have caused mutilations and deaths of passengers ejected from crashed vehicles. After an eighth reported death involving the safety handrails, MassDOT officials announced plans to cover or remove the allegedly dangerous fixtures, but only near curves or exit ramps. This partial removal of hazards has been criticized by a safety specialist, who suggests that the handrails are just as dangerous in straight sections of the tunnel.
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Lighting fixtures.
In March 2011, it became known that senior MassDOT officials had failed to disclose an issue with the lighting fixtures in the O'Neill tunnel. In early February 2011, a maintenance crew found a fixture lying in the middle travel lane in the northbound tunnel. Assuming it to be simple road debris, the maintenance team picked it up and brought it back to its home facility. The next day, a supervisor passing through the yard realized that the fixture was not road debris but was in fact one of the fixtures used to light the tunnel itself. Further investigation revealed that the fixture's mounting apparatus had failed, due to galvanic corrosion of incompatible metals, caused by having aluminum in direct contact with stainless steel, in the presence of salt water. The electrochemical potential difference between stainless steel and aluminum is in the range of 0.5 to 1.0V, depending on the exact alloys involved, and can cause considerable corrosion within months under unfavorable conditions.
After the discovery of the reason why the fixture had failed, a comprehensive inspection of the other fixtures in the tunnel revealed that numerous other fixtures were also in the same state of deterioration. Some of the worst fixtures were temporarily shored up with plastic ties. Moving forward with temporary repairs, members of the MassDOT administration team decided not to let the news of the systemic failure and repair of the fixtures be released to the public or to Governor Deval Patrick's administration.
, it appeared that all of the 25,000 light fixtures would have to be replaced, at an estimated cost of $54 million. The replacement work was mostly done at night, and required lane closures or occasional closing of the entire tunnel for safety, and was estimated to take up to two years to complete.
See also.
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Books of Chronicles
The Book of Chronicles ( , "words of the days") is a book in the Hebrew Bible, found as two books (1–2 Chronicles) in the Christian Old Testament. Chronicles is the final book of the Hebrew Bible, concluding the third section of the Jewish Tanakh, the Ketuvim ("Writings"). It contains a genealogy starting with Adam and a history of ancient Judah and Israel up to the Edict of Cyrus in 539 BC.
The book was translated into Greek and divided into two books in the Septuagint in the mid-3rd century BC. In Christian contexts Chronicles is referred to in the plural as the Books of Chronicles, after the Latin name given to the text by Jerome, but is also referred to by its Greek name as the Books of Paralipomenon. In Christian Bibles, they usually follow the two Books of Kings and precede Ezra–Nehemiah, the last history-oriented book of the Protestant Old Testament.
Summary.
The Chronicles narrative begins with Adam, Seth and Enosh, and the story is then carried forward, almost entirely through genealogical lists, down to the founding of the United Kingdom of Israel in the "introductory chapters", 1 Chronicles 1–9. The bulk of the remainder of 1 Chronicles, after a brief account of Saul in chapter 10, is concerned with the reign of David. The next long section concerns David's son Solomon, and the final part is concerned with the Kingdom of Judah, with occasional references to the northern Kingdom of Israel (2 Chronicles 10–36). The final chapter covers briefly the reigns of the last four kings, until Judah is destroyed and the people taken into exile in Babylon. In the two final verses, identical to the opening verses of the Book of Ezra, the Persian king Cyrus the Great conquers the Neo-Babylonian Empire, and authorises the restoration of the Temple in Jerusalem and the return of the exiles.
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Structure.
Originally a single work, Chronicles was divided into two in the Septuagint, a Greek translation produced in the 3rd and 2nd centuries BC. It has three broad divisions:
Within this broad structure there are signs that the author has used various other devices to structure his work, notably through drawing parallels between David and Solomon (the first becomes king, establishes the worship of Israel's God in Jerusalem, and fights the wars that will enable the Temple to be built, then Solomon becomes king, builds and dedicates the Temple, and reaps the benefits of prosperity and peace).
1 Chronicles is divided into 29 chapters and 2 Chronicles into 36 chapters. Biblical commentator C. J. Ball suggests that the division into two books introduced by the translators of the Septuagint "occurs in the most suitable place", namely with the conclusion of David's reign as king and the initiation of Solomon's reign.
The Talmud considered Chronicles one book.
Composition.
Origins.
The last events recorded in Chronicles take place in the reign of Cyrus the Great, the Persian king who conquered Babylon in 539 BC; this sets the earliest possible date for this passage of the book.
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Chronicles appears to be largely the work of a single individual. The writer was probably male, probably a Levite (temple priest), and probably from Jerusalem. He was well-read, a skilled editor, and a sophisticated theologian. He aimed to use the narratives in the Torah and former prophets to convey religious messages to his peers, the literary and political elite of Jerusalem in the time of the Achaemenid Empire.
Jewish and Christian tradition identified this author as the 5th-century BC figure Ezra, who gives his name to the Book of Ezra; Ezra is also believed by the Talmudic sages to have written both his own book (i. e., Ezra–Nehemiah) and Chronicles up to his own time, the latter having been finished by Nehemiah. Later critics, skeptical of the long-maintained tradition, preferred to call the author "the Chronicler". However, many scholars maintain support for Ezra's authorship, not only based on centuries of work by Jewish historians, but also due to the consistency of language and speech patterns between Chronicles and Ezra–Nehemiah. Professor Emeritus Menahem Haran of the Hebrew University of Jerusalem explains, "the overall unity of the Chronistic Work is … demonstrated by a common ideology, the uniformity of legal, cultic and historical conceptions and specific style, all of which reflect one opus."
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One of the most striking, although inconclusive, features of Chronicles is that its closing sentence is repeated as the opening of Ezra–Nehemiah. In antiquity, such repeated verses, like the "catch-lines" used by modern printers, often appeared at the end of a scroll to facilitate the reader's passing on to the correct second book-scroll after completing the first. This scribal device was employed in works that exceeded the scope of a single scroll and had to be continued on another scroll.
The latter half of the 20th century, amid growing skepticism in academia regarding history in the Biblical tradition, saw a reappraisal of the authorship question. Though there is a general lack of corroborating evidence, many now regard it as improbable that the author of Chronicles was also the author of the narrative portions of Ezra–Nehemiah. These critics suggest that "Chronicles" was probably composed between 400 and 250 BC, with the period 350–300 BC the most likely. This timeframe is achieved by estimates made based on genealogies appearing in the Greek Septuagint. This theory bases its premise on the latest person mentioned in Chronicles, Anani. Anani is an eighth-generation descendant of King Jehoiachin according to the Masoretic Text. This has persuaded many supporters of the Septuagint's reading to place Anani's likely date of birth a century later than what had been largely accepted for two millennia.
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Sources.
Much of the content of Chronicles is a repetition of material from other books of the Bible, from Genesis to Kings, and so the usual scholarly view is that these books, or an early version of them, provided the author with the bulk of his material. It is, however, possible that the situation was rather more complex, and that books such as Genesis and Samuel should be regarded as contemporary with Chronicles, drawing on much of the same material, rather than a source for it. Despite much discussion of this issue, no agreement has been reached. It is also likely that Chronicles preserved ancient heterodox traditions regarding Israel's history.
Genre.
The translators who created the Greek version of the Jewish Bible (the Septuagint) called this book "Paralipomenon", "Things Left Out", indicating that they thought of it as a supplement to another work, probably Genesis–Kings, but the idea seems inappropriate, since much of Genesis–Kings has been copied almost without change. Some modern scholars proposed that Chronicles is a midrash, or traditional Jewish commentary, on Genesis–Kings, but again this is not entirely accurate since the author or authors do not comment on the older books so much as use them to create a new work. Recent suggestions have been that it was intended as a clarification of the history in Genesis–Kings, or a replacement or alternative for it.
Themes.
Presbyterian theologian Paul K. Hooker argues that the generally accepted message the author wished to give to his audience was a theological reflection, not a "history of Israel":
External links.
Translations
Introductions
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Binary search tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is linear with respect to the height of the tree.
Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler.
The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of the binary search tree can be built with guaranteed worst-case performance. The basic operations include: search, traversal, insert and delete. BSTs with guaranteed worst-case complexities perform better than an unsorted array, which would require linear search time.
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The complexity analysis of BST shows that, on average, the insert, delete and search takes formula_1 for formula_2 nodes. In the worst case, they degrade to that of a singly linked list: formula_3. To address the boundless increase of the tree height with arbitrary insertions and deletions, self-balancing variants of BSTs are introduced to bound the worst lookup complexity to that of the binary logarithm. AVL trees were the first self-balancing binary search trees, invented in 1962 by Georgy Adelson-Velsky and Evgenii Landis.
Binary search trees can be used to implement abstract data types such as dynamic sets, lookup tables and priority queues, and used in sorting algorithms such as tree sort.
History.
The binary search tree algorithm was discovered independently by several researchers, including P.F. Windley, Andrew Donald Booth, Andrew Colin, Thomas N. Hibbard. The algorithm is attributed to Conway Berners-Lee and David Wheeler, who used it for storing labeled data in magnetic tapes in 1960. One of the earliest and popular binary search tree algorithm is that of Hibbard.
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The time complexity of a binary search tree increases boundlessly with the tree height if the nodes are inserted in an arbitrary order, therefore self-balancing binary search trees were introduced to bound the height of the tree to formula_1. Various height-balanced binary search trees were introduced to confine the tree height, such as AVL trees, Treaps, and red–black trees.
The AVL tree was invented by Georgy Adelson-Velsky and Evgenii Landis in 1962 for the efficient organization of information. It was the first self-balancing binary search tree to be invented.
Overview.
A binary search tree is a rooted binary tree in which nodes are arranged in strict total order in which the nodes with keys greater than any particular node "A" is stored on the right sub-trees to that node "A" and the nodes with keys equal to or less than "A" are stored on the left sub-trees to "A," satisfying the binary search property.
Binary search trees are also efficacious in sortings and search algorithms. However, the search complexity of a BST depends upon the order in which the nodes are inserted and deleted; since in worst case, successive operations in the binary search tree may lead to degeneracy and form a singly linked list (or "unbalanced tree") like structure, thus has the same worst-case complexity as a linked list.
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Binary search trees are also a fundamental data structure used in construction of abstract data structures such as sets, multisets, and associative arrays.
Operations.
Searching.
Searching in a binary search tree for a specific key can be programmed recursively or iteratively.
Searching begins by examining the root node. If the tree is , the key being searched for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and the node is returned. If the key is less than that of the root, the search proceeds by examining the left subtree. Similarly, if the key is greater than that of the root, the search proceeds by examining the right subtree. This process is repeated until the key is found or the remaining subtree is formula_5. If the searched key is not found after a formula_5 subtree is reached, then the key is not present in the tree.
Recursive search.
The following pseudocode implements the BST search procedure through recursion.
The recursive procedure continues until a formula_5 or the formula_8 being searched for are encountered.
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Iterative search.
The recursive version of the search can be "unrolled" into a while loop. On most machines, the iterative version is found to be more efficient.
Since the search may proceed till some leaf node, the running time complexity of BST search is formula_9 where formula_10 is the height of the tree. However, the worst case for BST search is formula_3 where formula_2 is the total number of nodes in the BST, because an unbalanced BST may degenerate to a linked list. However, if the BST is height-balanced the height is formula_1.
Successor and predecessor.
For certain operations, given a node formula_14, finding the successor or predecessor of formula_14 is crucial. Assuming all the keys of a BST are distinct, the successor of a node formula_14 in a BST is the node with the smallest key greater than formula_14's key. On the other hand, the predecessor of a node formula_14 in a BST is the node with the largest key smaller than formula_14's key. The following pseudocode finds the successor and predecessor of a node formula_14 in a BST.
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Operations such as finding a node in a BST whose key is the maximum or minimum are critical in certain operations, such as determining the successor and predecessor of nodes. Following is the pseudocode for the operations.
Insertion.
Operations such as insertion and deletion cause the BST representation to change dynamically. The data structure must be modified in such a way that the properties of BST continue to hold. New nodes are inserted as leaf nodes in the BST. Following is an iterative implementation of the insertion operation.
The procedure maintains a "trailing pointer" formula_21 as a parent of formula_14. After initialization on line 2, the while loop along lines 4-11 causes the pointers to be updated. If formula_21 is formula_5, the BST is empty, thus formula_25 is inserted as the root node of the binary search tree formula_26, if it is not formula_5, insertion proceeds by comparing the keys to that of formula_21 on the lines 15-19 and the node is inserted accordingly.
Deletion.
The deletion of a node, say formula_29, from the binary search tree formula_30 has three cases:
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The following pseudocode implements the deletion operation in a binary search tree.
The formula_51 procedure deals with the 3 special cases mentioned above. Lines 2-3 deal with case 1; lines 4-5 deal with case 2 and lines 6-16 for case 3. The helper function formula_52 is used within the deletion algorithm for the purpose of replacing the node formula_53 with formula_54 in the binary search tree formula_30. This procedure handles the deletion (and substitution) of formula_53 from formula_30.
Traversal.
A BST can be traversed through three basic algorithms: inorder, preorder, and postorder tree walks.
Following is a recursive implementation of the tree walks.
Balanced binary search trees.
Without rebalancing, insertions or deletions in a binary search tree may lead to degeneration, resulting in a height formula_2 of the tree (where formula_2 is number of items in a tree), so that the lookup performance is deteriorated to that of a linear search. Keeping the search tree balanced and height bounded by formula_1 is a key to the usefulness of the binary search tree. This can be achieved by "self-balancing" mechanisms during the updation operations to the tree designed to maintain the tree height to the binary logarithmic complexity.
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Height-balanced trees.
A tree is height-balanced if the heights of the left sub-tree and right sub-tree are guaranteed to be related by a constant factor. This property was introduced by the AVL tree and continued by the red–black tree. The heights of all the nodes on the path from the root to the modified leaf node have to be observed and possibly corrected on every insert and delete operation to the tree.
Weight-balanced trees.
In a weight-balanced tree, the criterion of a balanced tree is the number of leaves of the subtrees. The weights of the left and right subtrees differ at most by formula_61. However, the difference is bound by a ratio formula_62 of the weights, since a strong balance condition of formula_61 cannot be maintained with formula_1 rebalancing work during insert and delete operations. The formula_62-weight-balanced trees gives an entire family of balance conditions, where each left and right subtrees have each at least a fraction of formula_62 of the total weight of the subtree.
Types.
There are several self-balanced binary search trees, including T-tree, treap, red-black tree, B-tree, 2–3 tree, and Splay tree.
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Examples of applications.
Sort.
Binary search trees are used in sorting algorithms such as tree sort, where all the elements are inserted at once and the tree is traversed at an in-order fashion. BSTs are also used in quicksort.
Priority queue operations.
Binary search trees are used in implementing priority queues, using the node's key as priorities. Adding new elements to the queue follows the regular BST insertion operation but the removal operation depends on the type of priority queue: |
Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the "left child" and the "right child". That is, it is a "k"-ary tree with . A recursive definition using set theory is that a binary tree is a tuple ("L", "S", "R"), where "L" and "R" are binary trees or the empty set and "S" is a singleton set containing the root.
From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of "binary tree" to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted.
In mathematics, what is termed "binary tree" can vary significantly from author to author. Some use the definition commonly used in computer science, but others define it as every non-leaf having exactly two children and don't necessarily label the children as left and right either.
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In computing, binary trees can be used in two very different ways:
Definitions.
Recursive definition.
To define a binary tree, the possibility that only one of the children may be empty must be acknowledged. An artifact, which in some textbooks is called an "extended binary tree," is needed for that purpose. An extended binary tree is thus recursively defined as:
Another way of imagining this construction (and understanding the terminology) is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles.
Using graph theory concepts.
A binary tree is a rooted tree that is also an ordered tree (a.k.a. plane tree) in which every node has at most two children. A rooted tree naturally imparts a notion of levels (distance from the root); thus, for every node, a notion of children may be defined as the nodes connected to it a level below. Ordering of these children (e.g., by drawing them on a plane) makes it possible to distinguish a left child from a right child. But this still does not distinguish between a node with left but not a right child from a node with right but no left child.
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The necessary distinction can be made by first partitioning the edges; i.e., defining the binary tree as triplet (V, E1, E2), where (V, E1 ∪ E2) is a rooted tree (equivalently arborescence) and E1 ∩ E2 is empty, and also requiring that for all "j" ∈ { 1, 2 }, every node has at most one E"j" child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees.
Types of binary trees.
Tree terminology is not well-standardized and therefore may vary among examples in the available literature.
Combinatorics.
In combinatorics, one considers the problem of counting the number of full binary trees of a given size. Here the trees have no values attached to their nodes (this would just multiply the number of possible trees by an easily determined factor), and trees are distinguished only by their structure; however, the left and right child of any node are distinguished (if they are different trees, then interchanging them will produce a tree distinct from the original one). The size of the tree is taken to be the number "n" of internal nodes (those with two children); the other nodes are leaf nodes and there are of them. The number of such binary trees of size "n" is equal to the number of ways of fully parenthesizing a string of symbols (representing leaves) separated by "n" binary operators (representing internal nodes), to determine the argument subexpressions of each operator. For instance for one has to parenthesize a string like , which is possible in five ways:
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The correspondence to binary trees should be obvious, and the addition of redundant parentheses (around an already parenthesized expression or around the full expression) is disallowed (or at least not counted as producing a new possibility).
There is a unique binary tree of size 0 (consisting of a single leaf), and any other binary tree is characterized by the pair of its left and right children; if these have sizes "i" and "j" respectively, the full tree has size . Therefore, the number formula_30 of binary trees of size "n" has the following recursive description formula_31, and formula_32 for any positive integer "n". It follows that formula_30 is the Catalan number of index "n".
The above parenthesized strings should not be confused with the set of words of length 2"n" in the Dyck language, which consist only of parentheses in such a way that they are properly balanced. The number of such strings satisfies the same recursive description (each Dyck word of length 2"n" is determined by the Dyck subword enclosed by the initial '(' and its matching ')' together with the Dyck subword remaining after that closing parenthesis, whose lengths 2"i" and 2"j" satisfy ); this number is therefore also the Catalan number formula_30. So there are also five Dyck words of length 6:
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These Dyck words do not correspond to binary trees in the same way. Instead, they are related by the following recursively defined bijection: the Dyck word equal to the empty string corresponds to the binary tree of size 0 with only one leaf. Any other Dyck word can be written as (formula_35)formula_36, where formula_35,formula_36 are themselves (possibly empty) Dyck words and where the two written parentheses are matched. The bijection is then defined by letting the words formula_35 and formula_36 correspond to the binary trees that are the left and right children of the root.
A bijective correspondence can also be defined as follows: enclose the Dyck word in an extra pair of parentheses, so that the result can be interpreted as a Lisp list expression (with the empty list () as only occurring atom); then the dotted-pair expression for that proper list is a fully parenthesized expression (with NIL as symbol and '.' as operator) describing the corresponding binary tree (which is, in fact, the internal representation of the proper list).
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The ability to represent binary trees as strings of symbols and parentheses implies that binary trees can represent the elements of a free magma on a singleton set.
Methods for storing binary trees.
Binary trees can be constructed from programming language primitives in several ways.
Nodes and references.
In a language with records and references, binary trees are typically constructed by having a tree node structure which contains some data and references to its left child and its right child. Sometimes it also contains a reference to its unique parent. If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.
This method of storing binary trees wastes a fair bit of memory, as the pointers will be null (or point to the sentinel) more than half the time; a more conservative representation alternative is threaded binary tree.
In languages with tagged unions such as ML, a tree node is often a tagged union of two types of nodes, one of which is a 3-tuple of data, left child, and right child, and the other of which is a "leaf" node, which contains no data and functions much like the null value in a language with pointers. For example, the following line of code in OCaml (an ML dialect) defines a binary tree that stores a character in each node.
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type chr_tree = Empty | Node of char * chr_tree * chr_tree
Arrays.
Binary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree is a complete binary tree, this method wastes no space. In this compact arrangement, if a node has an index "i", its children are found at indices formula_41 (for the left child) and formula_42 (for the right), while its parent (if any) is found at index "formula_43" (assuming the root has index zero). Alternatively, with a 1-indexed array, the implementation is simplified with children found at formula_44 and formula_45, and parent found at formula_46.
This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal. It is often used for binary heaps.
Encodings.
Succinct encodings.
A succinct data structure is one which occupies close to minimum possible space, as established by information theoretical lower bounds. The number of different binary trees on formula_47 nodes is formula_48, the formula_47th Catalan number (assuming we view trees with identical "structure" as identical). For large formula_47, this is about formula_51; thus we need at least about formula_52 bits to encode it. A succinct binary tree therefore would occupy formula_53 bits.
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One simple representation which meets this bound is to visit the nodes of the tree in preorder, outputting "1" for an internal node and "0" for a leaf. If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder. This function accomplishes this:
function EncodeSuccinct("node" n, "bitstring" structure, "array" data) {
if n = "nil" then
append 0 to structure;
else
append 1 to structure;
append n.data to data;
EncodeSuccinct(n.left, structure, data);
EncodeSuccinct(n.right, structure, data);
The string "structure" has only formula_54 bits in the end, where formula_47 is the number of (internal) nodes; we don't even have to store its length. To show that no information is lost, we can convert the output back to the original tree like this:
function DecodeSuccinct("bitstring" structure, "array" data) {
remove first bit of "structure" and put it in "b"
if b = 1 then
create a new node "n"
remove first element of data and put it in n.data
n.left = DecodeSuccinct(structure, data)
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n.right = DecodeSuccinct(structure, data)
return n
else
return nil
More sophisticated succinct representations allow not only compact storage of trees but even useful operations on those trees directly while they're still in their succinct form.
Encoding ordered trees as binary trees.
There is a natural one-to-one correspondence between ordered trees and binary trees. It allows any ordered tree to be uniquely represented as a binary tree, and vice versa:
Let "T" be a node of an ordered tree, and let "B" denote "T's" image in the corresponding binary tree. Then "B's" "left" child represents "T's" first child, while the "B's right" child represents "T"'s next sibling.
For example, the ordered tree on the left and the binary tree on the right correspond:
In the pictured binary tree, the black, left, edges represent "first child", while the blue, right, edges represent "next sibling".
This representation is called a left-child right-sibling binary tree.
Common operations.
There are a variety of different operations that can be performed on binary trees. Some are mutator operations, while others simply return useful information about the tree.
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Insertion.
Nodes can be inserted into binary trees in between two other nodes or added after a leaf node. In binary trees, a node that is inserted is specified as to whose child it will be.
Leaf nodes.
To add a new node after leaf node A, A assigns the new node as one of its children and the new node assigns node A as its parent.
Internal nodes.
Insertion on internal nodes is slightly more complex than on leaf nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion.) A assigns its child to the new node and the new node assigns its parent to A. Then the new node assigns its child to B and B assigns its parent as the new node.
Deletion.
Deletion is the process whereby a node is removed from the tree. Only certain nodes in a binary tree can be removed unambiguously.
Node with zero or one children.
Suppose that the node to delete is node A. If A has no children, deletion is accomplished by setting the child of A's parent to null. If A has one child, set the parent of A's child to A's parent and set the child of A's parent to A's child.
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Node with two children.
In a binary tree, a node with two children cannot be deleted unambiguously. However, in certain binary trees (including binary search trees) these nodes "can" be deleted, though with a rearrangement of the tree structure.
Traversal.
Pre-order, in-order, and post-order traversal visit each node in a tree by recursively visiting each node in the left and right subtrees of the root. Below are the brief descriptions of above mentioned traversals.
Pre-order.
In pre-order, we always visit the current node; next, we recursively traverse the current node's left subtree, and then we recursively traverse the current node's right subtree. The pre-order traversal is a topologically sorted one, because a parent node is processed before any of its child nodes is done.
In-order.
In in-order, we always recursively traverse the current node's left subtree; next, we visit the current node, and lastly, we recursively traverse the current node's right subtree.
Post-order.
In post-order, we always recursively traverse the current node's left subtree; next, we recursively traverse the current node's right subtree and then visit the current node. Post-order traversal can be useful to get postfix expression of a binary expression tree.
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Depth-first order.
In depth-first order, we always attempt to visit the node farthest from the root node that we can, but with the caveat that it must be a child of a node we have already visited. Unlike a depth-first search on graphs, there is no need to remember all the nodes we have visited, because a tree cannot contain cycles. Pre-order is a special case of this. See depth-first search for more information.
Breadth-first order.
Contrasting with depth-first order is breadth-first order, which always attempts to visit the node closest to the root that it has not already visited. See breadth-first search for more information. Also called a "level-order traversal".
In a complete binary tree, a node's breadth-index ("i" − (2"d" − 1)) can be used as traversal instructions from the root. Reading bitwise from left to right, starting at bit "d" − 1, where "d" is the node's distance from the root ("d" = ⌊log("i"+1)⌋) and the node in question is not the root itself ("d" > 0). When the breadth-index is masked at bit "d" − 1, the bit values and mean to step either left or right, respectively. The process continues by successively checking the next bit to the right until there are no more. The rightmost bit indicates the final traversal from the desired node's parent to the node itself. There is a time-space trade-off between iterating a complete binary tree this way versus each node having pointer(s) to its sibling(s). |
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
Formal definition.
Let formula_1 be a locally compact Hausdorff space, and let formula_2 be the smallest σ-algebra that contains the open sets of formula_1; this is known as the σ-algebra of Borel sets. A Borel measure is any measure formula_4 defined on the σ-algebra of Borel sets. A few authors require in addition that formula_4 is locally finite, meaning that every point has an open neighborhood with finite measure. For Hausdorff spaces, this implies that formula_6 for every compact set formula_7; and for locally compact Hausdorff spaces, the two conditions are equivalent. If a Borel measure formula_4 is both inner regular and outer regular, it is called a regular Borel measure. If formula_4 is both inner regular, outer regular, and locally finite, it is called a Radon measure. Alternatively, if a regular Borel measure formula_4 is tight, it is a Radon measure.
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If formula_1 is a separable complete metric space, then every Borel measure formula_4 on formula_1 is a Radon measure.
On the real line.
The real line formula_14 with its usual topology is a locally compact Hausdorff space; hence we can define a Borel measure on it. In this case, formula_15 is the smallest σ-algebra that contains the open intervals of formula_14. While there are many Borel measures "μ", the choice of Borel measure that assigns formula_17 for every half-open interval formula_18 is sometimes called "the" Borel measure on formula_14. This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure formula_20, which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the "completion" of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and can be equipped with a complete measure. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., formula_21 for every Borel measurable set, where formula_4 is the Borel measure described above). This idea extends to finite-dimensional spaces formula_23 (the Cramér–Wold theorem, below) but does not hold, in general, for infinite-dimensional spaces. Infinite-dimensional Lebesgue measures do not exist.
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Product spaces.
If "X" and "Y" are second-countable, Hausdorff topological spaces, then the set of Borel subsets formula_24 of their product coincides with the product of the sets formula_25 of Borel subsets of "X" and "Y". That is, the Borel functor
from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
Applications.
Lebesgue–Stieltjes integral.
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.
Laplace transform.
One can define the Laplace transform of a finite Borel measure "μ" on the real line by the Lebesgue integral
An important special case is where "μ" is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function "f". In that case, to avoid potential confusion, one often writes
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where the lower limit of 0− is shorthand notation for
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.
Moment problem.
One can define the moments of a finite Borel measure "μ" on the real line by the integral
For formula_31 these correspond to the Hamburger moment problem, the Stieltjes moment problem and the Hausdorff moment problem, respectively. The question or problem to be solved is, given a collection of such moments, is there a corresponding measure? For the Hausdorff moment problem, the corresponding measure is unique. For the other variants, in general, there are an infinite number of distinct measures that give the same moments.
Hausdorff dimension and Frostman's lemma.
Given a Borel measure "μ" on a metric space "X" such that "μ"("X") > 0 and "μ"("B"("x", "r")) ≤ "rs" holds for some constant "s" > 0 and for every ball "B"("x", "r") in "X", then the Hausdorff dimension dimHaus("X") ≥ "s". A partial converse is provided by the Frostman lemma:
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Lemma: Let "A" be a Borel subset of R"n", and let "s" > 0. Then the following are equivalent:
Cramér–Wold theorem.
The Cramér–Wold theorem in measure theory states that a Borel probability measure on formula_33 is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold. |
Blackadder
Blackadder is a series of four period British sitcoms, plus several one-off instalments, which originally aired on BBC1 from 1983 to 1989. All television episodes starred Rowan Atkinson as the antihero Edmund Blackadder and Tony Robinson as Blackadder's dogsbody, Baldrick. Each series was set in a different historical period, with the two protagonists accompanied by different characters, though several reappear in one series or another, e.g., Melchett (Stephen Fry), Lord Percy Percy / Captain Darling (Tim McInnerny) and George (Hugh Laurie).
The first series, "The Black Adder", was written by Richard Curtis and Atkinson, while subsequent series were written by Curtis and Ben Elton. The shows were produced by John Lloyd. In 2000, the fourth series, "Blackadder Goes Forth", ranked at 16 in the 100 Greatest British Television Programmes, a list created by the British Film Institute. In a 2001 poll by Channel 4, Edmund Blackadder was ranked third on their list of the 100 Greatest TV Characters. In the 2004 TV poll to find Britain's Best Sitcom, "Blackadder" was voted the second-best British sitcom of all time, topped by "Only Fools and Horses". It was also ranked as the 9th-best TV show of all time by "Empire" magazine. Atkinson said "Blackadder" is "the thing he found the least stressful" to do.
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Premise.
Each series comprises six half-hour episodes and is set in a different period of British history. The first series, made in 1983, was titled "The Black Adder" and was set in the fictional reign of "Richard IV". The second series, "Blackadder II" (1986), was set during the reign of Elizabeth I. "Blackadder the Third" (1987) was set in the late 18th and early 19th centuries during the reign of George III. "Blackadder Goes Forth" (1989) was set in 1917 in the trenches of the Great War.
"Blackadder" follows the misfortunes of Edmund Blackadder (played by Atkinson). It is implied in each series that the Blackadder character is a descendant of the previous one. The end theme lyrics of the series 2 episode "Head" specify that he is the great-grandson of the previous incarnation, although it is never specified how or when any of the Blackadders (who are usually bachelors) manage to father children.
In series one, Edmund Blackadder is not particularly bright, and is much the intellectual inferior of his servant, Baldrick (played by Tony Robinson). However, in subsequent series, the positions are reversed: Blackadder is clever, shrewd, scheming and manipulative while Baldrick is extremely dim.
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Each incarnation of Blackadder and Baldrick is also saddled with tolerating the presence of a dimwitted aristocrat. In the first two series, this is Lord Percy Percy, played by Tim McInnerny. Hugh Laurie plays the role in the third and fourth series, as Prince George, Prince Regent, and Lieutenant George, respectively. Stephen Fry's Lord Melchett fills a similar role in the second and fourth series.
Episodes.
Series 1: "The Black Adder".
"The Black Adder", the first series of "Blackadder", was written by Richard Curtis and Rowan Atkinson and produced by John Lloyd. It originally aired on BBC1 from 15 June 1983 to 20 July 1983, and was a joint production with the Australian Seven Network.
Set in 1485 at the end of the British Middle Ages, the series is written as an alternative history in which Richard III won the Battle of Bosworth Field only to be mistaken for someone else and murdered, and is succeeded by Richard IV (Brian Blessed), one of the Princes in the Tower. The series follows the exploits of Richard IV's unfavoured second son Edmund, the Duke of Edinburgh (who calls himself "The Black Adder") in his various attempts to increase his standing with his father and his eventual quest to overthrow him. Guest appearances in this series include Peter Cook as King Richard III, Russell Enoch as the Duke of Winchester, Miriam Margolyes as the Infanta Maria Escalosa of Spain (with Jim Broadbent as her interpreter), Frank Finlay as the Witchsmeller Pursuivant, Valentine Dyall as Lord Angus, Stephen Frost and Mark Arden as guards, and Rik Mayall as Mad Gerald.
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Conceived while Atkinson and Curtis were working on "Not the Nine O'Clock News", the series dealt comically with a number of aspects of medieval life in Britain: witchcraft, royal succession, European relations, the Crusades, and the conflict between the Church and the Crown. Along with the secret history, many historical events portrayed in the series were anachronistic (for example, Constantinople had already fallen to the Ottoman Empire in 1453, predating the events in the episode by 32 years); this dramatic license would continue in the subsequent "Blackadders". The filming of the series was highly ambitious, with a large cast and much location shooting. The series also featured Shakespearean dialogue, often adapted for comic effect; the end credits featured the words "Additional Dialogue by William Shakespeare".
Series 2: "Blackadder II".
"Blackadder II" is set in England during the reign of Queen Elizabeth I (1558–1603), who is portrayed by Miranda Richardson. The principal character is Edmund, Lord Blackadder, the great-grandson of the original Black Adder. During the series, he regularly deals with the Queen, her obsequious Lord Chamberlain Lord Melchett (Stephen Fry; his rival for the Queen's affections), his friend Lord Percy Percy (played by Tim McInnerny) and the Queen's demented former nanny Nursie (Patsy Byrne). Guest appearances in the series include Tom Baker as Captain Redbeard Rum, Simon Jones as Sir Walter Raleigh, Ronald Lacey as the Bishop of Bath and Wells, and Miriam Margoyles as Blackadder's aunt, Lady Whiteadder. The series also features two appearances by Hugh Laurie (as Simon Partridge, a friend of Blackadder's, in the episode "Beer"; and as Prince Ludwig the Indestructible in the series' finale "Chains"), as well as the first appearance of Gabrielle Glaister as "Bob", and of Rik Mayall as Lord Flashheart.
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Following the BBC's request for improvements (and a severe budget reduction), several changes were made. The second series was the first to establish the familiar Blackadder character: cunning, shrewd and witty, in sharp contrast to the first series' bumbling Prince Edmund. To reduce the cost of production, it was shot with virtually no outdoor scenes (the first series was shot largely on location) and several frequently used indoor sets, such as the Queen's throne room and Blackadder's front room.
A quote from this series ranked number three in a list of the top 25 television "putdowns" of the last 40 years by the "Radio Times" magazine: "The eyes are open, the mouth moves, but Mr. Brain has long since departed, hasn't he, Percy?"
Series 3: "Blackadder the Third".
"Blackadder the Third" is set in the late 18th and early 19th centuries, a period known as the Regency. In the series, Edmund Blackadder Esquire is a butler to George IV, who is played as a buffoonish fop. Despite Edmund's respected intelligence and abilities, he has no personal fortune to speak of, apart from his frequently fluctuating wage packet from the Prince (“If I’m running short of cash, all I have to do is go upstairs and ask Prince Fathead for a raise”), and from (it seems) stealing the Prince's socks and selling them off. The episode titles were puns on Jane Austen’s novels "Sense and Sensibility" and "Pride and Prejudice".
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Along with Rowan Atkinson and Tony Robinson in their usual roles, this series starred Hugh Laurie as the Prince Regent and Helen Atkinson-Wood as Mrs. Miggins. The series features Dr. Samuel Johnson (Robbie Coltrane); William Pitt the Younger (Simon Osborne); the French Revolution (with Chris Barrie, Tim McInnerny as the Scarlet Pimpernel, and Nigel Planer); hammy theatrical actors (Kenneth Connor and Hugh Paddick); a squirrel-hating cross-dressing highwayman (Miranda Richardson); and a duel with the Duke of Wellington (Stephen Fry).
Series 4: "Blackadder Goes Forth".
This series is set in 1917, on the Western Front of the First World War. Another "big push" is planned, and Captain Blackadder's one goal is to avoid being killed, but his schemes always land him back in the trenches. Blackadder is joined by his batman Private S. Baldrick (Tony Robinson) and idealistic Edwardian twit Lieutenant George (Hugh Laurie). General Melchett (Stephen Fry) rallies his troops from a French château from the front, where he is aided and abetted by his assistant, Captain Kevin Darling (Tim McInnerny), pencil-pusher supreme and Blackadder's nemesis, whose name is played on for maximum comedic value. Guest appearances in this series include Stephen Frost as the leader of a firing squad detail, Miranda Richardson as Nurse Mary Fletcher-Brown, two further appearances of Gabrielle Glaister as "Bob" (in this series, a young woman who pretended to be a boy in order to join the army), Rik Mayall appearing as Royal Flying Corps Squadron Commander The Lord Flasheart, Adrian Edmondson as Baron Manfred von Richthofen (aka "The Red Baron"), and Geoffrey Palmer as Field Marshal Douglas Haig.
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The series' tone is somewhat darker than the other "Blackadder"s; it details the privations of trench warfare as well as the incompetence and life-wasting strategies of the top brass. For example, Baldrick is reduced to cooking rats and making coffee from mud, while General Melchett hatches a plan for the troops to walk very slowly toward the German lines, because "it'll be the last thing Fritz will expect."
The final episode, "Goodbyeee", is known for being extraordinarily poignant for a comedy – especially the final scene, which sees the main characters (Blackadder, Baldrick, George, and Darling) finally going "over the top" and charging off into the fog and smoke of no man's land, presumably to die. In a list of the 100 Greatest British Television Programmes, drawn up by the British Film Institute in 2000 and voted for by industry professionals, "Blackadder Goes Forth" was placed 16th.
Specials.
Pilot episode.
The "Blackadder" pilot was shot but never broadcast on TV in the UK (although some scenes were shown in the 25th anniversary special "Blackadder Rides Again"). One notable difference in the pilot, as in many pilots, is the casting. Baldrick is played not by Tony Robinson, but by Philip Fox. Another significant difference is that the character of Prince Edmund presented in the pilot is much closer to the intelligent, conniving Blackadder of the later series than the snivelling, weak buffoon of the original. Set in the year 1582, the script of the pilot is roughly the same as the episode "Born to Be King", albeit with some different jokes, with some lines appearing in other episodes of the series.
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UKTV Gold broadcast the pilot on 15 June 2023, as part of an 80-minute special hosted by Sir Tony Robinson and featuring interviews with Ben Elton and Richard Curtis.
"Blackadder: The Cavalier Years".
This special, set in the English Civil War, was shown as part of Comic Relief's Red Nose Day on Friday 5 February 1988. The 15-minute episode is set in November 1648, during the last days of the Civil War. Sir Edmund Blackadder and his servant, Baldrick, are the last two men loyal to the defeated King Charles I of England (played by Stephen Fry), portrayed as a soft-spoken, ineffective, naive character, with the voice and mannerisms of Charles I's namesake, the then Prince of Wales (now Charles III). However, owing to a misunderstanding between Oliver Cromwell (guest-star Warren Clarke) and Baldrick, the King is arrested and sent to the Tower of London. The rest of the episode revolves around Blackadder's attempts to save the King as well as improve his own standing.
"Blackadder's Christmas Carol".
The second special was broadcast on Friday 23 December 1988. In a twist on Charles Dickens' "A Christmas Carol", Ebenezer Blackadder is the "kindest and loveliest" man in England. The Spirit of Christmas shows Blackadder the contrary antics of his ancestors and descendants, and reluctantly informs him that if he turns evil his descendants will enjoy power and fortune, while if he remains the same a future Blackadder will live shamefully subjugated to a future incompetent Baldrick. This remarkable encounter causes him to proclaim, "Bad guys have all the fun", and adopt the personality with which viewers are more familiar.
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"Blackadder: Back & Forth".
"Blackadder: Back & Forth" was originally shown in the Millennium Dome in 2000, followed by a screening on Sky One in the same year (and later on BBC1). It is set on the turn of the millennium, and features Lord Blackadder placing a bet with his friends – modern versions of Queenie (Miranda Richardson), Melchett (Stephen Fry), George (Hugh Laurie) and Darling (Tim McInnerny) – that he has built a working time machine. While this is intended as a clever con trick, the machine surprisingly works, sending Blackadder and Baldrick back to the Cretaceous period, where they manage to cause the extinction of the dinosaurs through the use of Baldrick's best-worst-and-only pair of underpants as a weapon against a hungry T. Rex. Finding that Baldrick has forgotten to write dates on the machine's dials, the rest of the film follows their attempts to find their way back to 1999, often creating huge historical anomalies in the process that must be corrected before the end. The film includes cameo appearances from Kate Moss and Colin Firth.
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"The Big Night In".
Broadcast in 2020 as part of Children in Need and Comic Relief's joint special "The Big Night In" during the COVID-19 pandemic, Fry resumed the role of Lord Melchett (an intellectually-brilliant version), Head of the Royal Household, under lockdown at Melchett Manor, to help Prince William deal with educating his children via Zoom and discussing "Tiger King", before they both step outside to clap for the National Health Service. Melchett is said to be isolating with Lord Blackadder, both grandsons to their First World War counterparts.
Live stage performances.
In 1998, as part of Prince Charles' 50th Birthday Gala televised on ITV, Atkinson appeared as a Restoration Blackadder reading aloud a letter to the Privy Council of King Charles II. He colourfully refuses their invitation to stage a royal gala, calling such occasions "very, very, very dull" and asserting that there was "more musical talent on display when my servant Baldrick breaks wind."
In 2000, on the BBC's annual Royal Variety Performance, Atkinson portrayed Blackadder as a present-day officer in "Her Majesty's Royal Regiment of Shirkers" and delivered a monologue titled "Blackadder: The Army Years", proposing that Britain regain her former greatness by invading (or at least buying) France.
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In 2012, as part of the Prince's Trust charity show "We Are Most Amused", Atkinson and Robinson reprised their roles as Blackadder and Baldrick in a comedy sketch featuring Miranda Hart as leader of a government inquiry into the recent banking crisis. Blackadder, chief executive of a fictional British bank, appearing with Baldrick as his gardener, convinces the panel to publicly blame the entire crisis on Baldrick, to the latter's consternation.
Red Nose Day 2023.
Baldrick (Tony Robinson) returned in 2023 for a Red Nose Day sketch for the BBC. There was no involvement of Rowan Atkinson or a subsequent reboot, amid speculation.
Production.
Series development.
Rowan Atkinson and Richard Curtis developed the idea for the sitcom while working on "Not the Nine O'Clock News". Eager to avoid comparisons to the critically acclaimed "Fawlty Towers", they proposed the idea of a historical sitcom. A pilot episode was made in 1982, and a six-episode series was commissioned. The budget for the series was considerable, with much location shooting particularly at Alnwick Castle in Northumberland and the surrounding countryside in February 1983. The series also used large casts of extras, horses and expensive medieval-style costumes. Atkinson has said about the making of the first series:
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The first series was odd, it was very extravagant. It cost a million pounds for the six programmes ... [which] was a lot of money to spend ... It looked great, but it wasn't as consistently funny as we would have liked.
Owing to the high cost of the first series, the then-controller of programming of BBC1, Michael Grade, was reluctant to sign off a second series without major improvements to the show and drastic cost-cutting, leaving a gap of three years between the two series.
A chance meeting between Richard Curtis and comedian Ben Elton led to the decision to collaborate on a new series of Blackadder. Recognising the main faults of the first series, Curtis and Elton agreed that "Blackadder II" would be a studio-only production (along with the inclusion of a live audience during recording, instead of showing the episodes to an audience after taping). Besides adding a greater comedy focus, Elton suggested a major change in character emphasis: Baldrick would become the stupid sidekick, while Edmund Blackadder evolved into a cunning sycophant. This led to the familiar set-up that was maintained in the following series.
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Only in the "Back & Forth" millennium special was the shooting once again on location, because this was a production with a budget estimated at £3 million, and was a joint venture between Tiger Aspect, Sky Television, the New Millennium Experience Company and the BBC, rather than the BBC alone.
Casting.
Each series tended to feature the same set of regular actors in different period settings, although throughout the four series and specials, only Blackadder and Baldrick were constant characters. Several regular cast members recurred as characters with similar names, implying, like Blackadder, that they were descendants.
Recurring cast.
Various actors have appeared in more than one of the Blackadder series and/or specials. These are:
Guest cast.
Ben Elton's arrival after the first series heralded the more frequent recruitment of comic actors from the alternative comedy era for guest appearances, including Robbie Coltrane, Rik Mayall (who had appeared in the final episode of the first series as "Mad Gerald"), Adrian Edmondson, Nigel Planer, Mark Arden, Stephen Frost, Chris Barrie and Jeremy Hardy. Elton himself played an anarchist in "Blackadder the Third".
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Gabrielle Glaister played Bob, an attractive girl who poses as a man, in both series 2 and Driver Parkhurst in series 4. Rik Mayall plays Lord Flashheart, a vulgar friend in his first appearance and then a successful rival of Blackadder in later episodes of series 2 and 4. He also played a decidedly Flashheart-like Robin Hood in "Back & Forth". Lee Cornes also appeared in an episode of all three Curtis-Elton series. He appeared as a guard in the episode "Chains" of "Blackadder II"; as the poet Shelley in the episode "Ink and Incapability' of "Blackadder the Third"; and as firing squad soldier Private Fraser in the episode "Corporal Punishment" of "Blackadder Goes Forth".
More established actors, some at the veteran stage of their careers, were also recruited for roles. These included Peter Cook, John Grillo, Simon Jones, Tom Baker, Jim Broadbent, Hugh Paddick, Frank Finlay, Kenneth Connor, Bill Wallis, Ronald Lacey, Roger Blake, Denis Lill, Warren Clarke and Geoffrey Palmer, who played Field Marshal Sir Douglas Haig in "Goodbyeee", the final episode of "Blackadder Goes Forth". Miriam Margolyes played three different guest roles: The Spanish Infanta in The Queen of Spain's Beard, Lady Whiteadder in Beer, and Queen Victoria in Blackadder's Christmas Carol.
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Unusually for a sitcom based loosely on factual events and in the historical past, a man was recruited for one episode essentially to play himself. Political commentator Vincent Hanna played a character billed as "his own great-great-great grandfather" in the episode "Dish and Dishonesty" of "Blackadder the Third". Hanna was asked to take part because the scene was of a by-election in which Baldrick was a candidate and, in the style of modern television, Hanna gave a long-running "live" commentary of events at the count (and interviewed candidates and election agents) to a crowd through the town hall window.
Theme tune.
Howard Goodall's theme tune has the same melody throughout all the series, but is played in roughly the style of the period in which it is set. It is performed mostly with trumpets and timpani in "The Black Adder", the fanfares used suggesting typical medieval court fanfares; with a combination of recorder, string quartet and electric guitar in "Blackadder II" (the end theme, with different lyrics each time reflecting on the episode's events, was sung by a countertenor); on oboe, cello and harpsichord (in the style of a minuet) for "Blackadder the Third"; by The Band of the 3rd Battalion, Royal Anglian Regiment in "Blackadder Goes Forth"; sung by carol singers in "Blackadder's Christmas Carol"; and by an orchestra in "Blackadder: The Cavalier Years" and "Blackadder: Back & Forth".
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Awards.
In 2000, the fourth series, "Blackadder Goes Forth", ranked at 16 in the "100 Greatest British Television Programmes", a list created by the British Film Institute. In 2004, a BBC TV poll for "Britain's Best Sitcom", "Blackadder" was voted the second best British sitcom of all time, topped by "Only Fools and Horses". It was also ranked as the 20th Best TV Show of All Time by "Empire" magazine.
Future.
Despite regular statements denying any plans for a fifth series, cast members are regularly asked about the possibility of a new series.
In January 2005, Tony Robinson told ITV's "This Morning" that Rowan Atkinson was more keen than he has been in the past to do a fifth series, set in the 1960s (centred on a rock band called the "Black Adder Five", with Baldrick – a.k.a. 'Bald Rick' – as the drummer). In the documentary "Blackadder Rides Again", Robinson stated that the series would present Blackadder as the bastard son of Queen Elizabeth II and running a Beatles-like rock band. Rowan Atkinson, Tony Robinson, Hugh Laurie, Stephen Fry, Tim McInnerny and Miranda Richardson would have reprised their roles, and reportedly, Brian Blessed, Elspet Gray and Robert East would have returned from the first series to play Blackadder's biological family. Robinson in a stage performance 1 June 2007, again mentioned this idea, but in the context of a movie.
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One idea mentioned by Curtis was that it was Baldrick who had accidentally assassinated John F. Kennedy. However, aside from a brief mention in June 2005,
there have been no further announcements from the BBC that a new series is being planned. Furthermore, in November 2005, Rowan Atkinson told "BBC Breakfast" that, although he would very much like to do a new series set in Colditz or another prisoner-of-war camp during World War II, something which both he and Stephen Fry reiterated at the end of "Blackadder Rides Again", the chances of it happening are extremely slim.
There were a couple of ideas that had previously floated for the fifth series. "Batadder" was intended to be a parody of "Batman" with Baldrick as the counterpart of Robin (suggested by John Lloyd). This idea eventually came to surface as part of the "Comic Relief" sketch "Spider-Plant Man" in 2005, with Atkinson as the title hero, Robinson as Robin, Jim Broadbent as Batman and Rachel Stevens as Mary Jane. "Star Adder" was to be set in space in the future (suggested by Atkinson).
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On 10 April 2007, "Hello!" reported that Atkinson was moving forward with his ideas for a fifth series. He said, "I like the idea of him being a prisoner of war in Colditz. That would have the right level of authority and hierarchy which is apparent in all the Blackadders."
Stephen Fry has expressed the view that, since the series went out on such a good "high", a film might not be a good idea.
During his June 2007 stage performance, chronicled on the Tony Robinson's Cunning Night Out DVD, Robinson states that, after filming the "Back & Forth special", the general idea was to reunite for another special in 2010. Robinson jokingly remarked that Hugh Laurie's success on "House" may make that difficult.
On 28 November 2012, Rowan Atkinson reprised the role at the "We are most amused" comedy gala for the Prince's Trust at the Royal Albert Hall. He was joined by Tony Robinson as Baldrick. The sketch involved Blackadder as CEO of Melchett, Melchett and Darling bank facing an enquiry over the banking crisis.
In August 2015, Tony Robinson said in an interview "I do think a new series of Blackadder is on the cards. I have spoken to virtually all the cast about this now. The only problem is Hugh [Laurie]'s fee. He's a huge star now." However, in October 2018, Richard Curtis "dashed hopes" that the show would return for a fifth series.
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In April 2017 at the BFI & Radio Times Television Festival, Atkinson stated "There are no plans to do anything" and revealed a potential Russian Revolution themed series that never materialised:
"There was a plan twenty years ago that got nowhere which was called "Redadder" which I quite liked. It was set in Russia in 1917 and "Blackadder" and Baldrick were working for the Tsar. They had blue stripes around their caps and then the Revolution happened and Rik Mayall unsurprisingly was playing Rasputin."
In December 2020, Rowan Atkinson told the "Radio Times":
"I don't actually like the process of making anything – with the possible exception of "Blackadder". Because the responsibility for making that series funny was on many shoulders, not just mine. "Blackadder" represented the creative energy we all had in the '80s. To try to replicate that 30 years on wouldn't be easy."
Most recently, in December 2024, Ben Elton poured doubt on a fifth series of "Blackadder":
"But there will not be a fifth series of "Blackadder", I think that’s pretty much a certainty. I have no interest in doing it. I don’t think any of us do, with the possible exception of Tony [Robinson]. But if we did, the world would be our oyster. We could have fun with any period."
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Home media.
All series and many of the specials are available on VHS tapes, DVD & Blu-ray. Many are also available on BBC audio cassette. As of 2008, a "Best of BBC" edition box set is available containing all four major series together with "Blackadder's Christmas Carol" and "Back & Forth". All four series and the Christmas special are also available for download on iTunes.
VHS releases.
On 5 February 1990, BBC Enterprises Ltd released the first series on two single VHS tapes.
On 2 October 1989, BBC Enterprises Ltd released the second series on two single VHS tapes.
On 6 February 1989, BBC Enterprises Ltd released the third series on two single VHS tapes.
On 10 September 1990, BBC Enterprises Ltd released the fourth and final series on two single VHS tapes.
On 7 September 1992, all eight single Blackadder video releases were re-released as four "complete" double VHS releases. The four entire series videos were re-released as single VHS tape releases on 2 October 1995.
On 5 January 1998, five episodes of the first two series were released on a 15-rated VHS tape compilation by BBC Worldwide Ltd.
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On 4 November 1991, "Blackadder's Christmas Carol" was released on a single VHS tape release rated PG (Cat. No. BBCV 4646).
LP box set.
On 19 October 2022 there was an announcement that there will be a LP box set release and collects the Blackadder soundtracks on vinyl for the first time.
The deluxe 12-disc LP collection with the title "Blackadder's Historical Record" was pressed on gold-coloured 140g vinyl, and released on 10 February 2023 by Demon Records. It also includes a frameable print of Baldrick, each hand signed by Sir Tony Robinson himself and a comprehensive full-colour booklet detailing the comedy series, the "leather-look rigid box"
Stamps.
Royal Mail issued a set of special stamps celebrating "Blackadder" on 17 May 2023. |
Boii
The Boii (Latin plural, singular "Boius"; ) were a Celtic tribe of the later Iron Age, attested at various times in Cisalpine Gaul (present-day Northern Italy), Pannonia (present-day Austria and Hungary), present-day Bavaria, in and around present-day Bohemia (after whom the region is named in most languages; comprising the bulk of today's Czech Republic), parts of present-day Slovakia and Poland, and Gallia Narbonensis (located in modern Languedoc and Provence).
In addition, the archaeological evidence indicates that in the 2nd century BC Celts expanded from Bohemia through the Kłodzko Valley into Silesia, now part of Poland and the Czech Republic.
They first appear in history in connection with the Gallic invasion of northern Italy, 390 BC, when they made the Etruscan city of Felsina their new capital, Bononia (Bologna).
After a series of wars, they were decisively beaten by the Romans in the Battle of Mutina (193 BC) and their territory became part of the Roman province of Cisalpine Gaul. According to Strabo, writing two centuries after the events, rather than being destroyed by the Romans like their Celtic neighbours,
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Around 60 BC, a group of Boii joined the Helvetiis' ill-fated attempt to conquer land in western Gaul and were defeated by Julius Caesar, along with their allies, in the Battle of Bibracte.
Caesar settled the remnants of that group in Gorgobina, from where they sent 2,000 warriors to Vercingetorix's aid at the Battle of Alesia six years later. The eastern Boii on the Danube were incorporated into the Roman Empire in 8 AD.
Etymology and name.
From all the different names of the same Celtic people in literature and inscriptions, it is possible to abstract a Continental Celtic segment, .
There are two major derivations of this segment, both presupposing that it belongs to the family of Indo-European languages: from 'cow' and from 'warrior.' The Boii would thus be either 'the herding people' or 'the warrior people'.
The 'cow' derivation depends most immediately on the Old Irish legal term for 'outsider': "ambue", from Proto-Celtic (<"*an-bouios"), 'not a cattle owner'.
In a reference to the first known historical Boii, Polybius relates that their wealth consisted of cattle and gold, that they depended on agriculture and war, and that a man's status depended on the number of associates and assistants he had. The latter were presumably the ', as opposed to the man of status, who was ', a cattle owner, and the "" were originally a class, 'the cattle owners'.
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The 'warrior' derivation was adopted by the linguist Julius Pokorny, who presented it as being from Indo-European , , 'hit'; however, not finding any Celtic names close to it (except for the Boii), he adduces examples somewhat more widely from originals further back in time: "phohiio-s-", a Venetic personal name; "Boioi", an Illyrian tribe; "Boiōtoi", a Greek tribal name (the Boeotians); and a few others.
The same wider connections can be hypothesized for the 'cow' derivation: the Boeotians have been known for well over a century as a people of kine, which might have been parallel to the meaning of Italy as 'land of calves'. Indo-European reconstructions can be made using 'cow' as a basis, such as ; the root may itself be an imitation of the sound a cow makes.
Contemporary derived words include "Boiorix" ('king of the Boii', one of the chieftains of the Cimbri) and "Boiodurum" ('gate/fort of the Boii', modern Passau) in Germany. Their memory also survives in the modern regional names of Bohemia ("Boiohaemum"), a mixed-language form from and Proto-Germanic , 'home': 'home of the Boii', and , Bavaria, which is derived from the Germanic "Baiovarii" tribe (Germanic "*baja-warjaz": the first component is most plausibly explained as a Germanic version of "Boii"; the second part is a common formational morpheme of Germanic tribal names, meaning 'dwellers', as in Old English "-ware"); this combination 'Boii-dwellers' may have meant 'those who dwell where the Boii formerly dwelt'.
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History.
Settlement in north Italy.
According to the ancient authors, the Boii arrived in northern Italy by crossing the Alps. While of the other tribes who had come to Italy along with the Boii, the Senones, Lingones and Cenomani are also attested in Gaul at the time of the Roman conquest. It remains therefore unclear where exactly the Central Europe origins of the Boii lay, if somewhere in Gaul, Southern Germany or in Bohemia.
Polybius relates that the Celts were close neighbors of the Etruscan civilization and "cast covetous eyes on their beautiful country".
Invading the Po Valley with a large army, they drove out the Etruscans and resettled it, the Boii taking the right bank in the center of the valley. Strabo confirms that the Boii emigrated from their lands across the Alps and were one of the largest tribes of the Celts. The Boii occupied the old Etruscan settlement of Felsina, which they named "Bononia" (modern Bologna). Polybius describes the Celtic way of life in Cisalpine Gaul as follows:
The archaeological evidence from Bologna and its vicinity contradicts the testimony of Polybius and Livy on some points, who say the Boii expelled the Etruscans and perhaps some were forced to leave.
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It indicates the Boii neither destroyed nor depopulated Felsinum, but simply moved in and became part of the population by intermarriage.
The cemeteries of the period in Bologna contain La Tène weapons and other artifacts, as well as Etruscan items such as bronze mirrors. At Monte Bibele not far away one grave contained La Tène weapons and a pot with an Etruscan female name scratched on it.
War against Rome.
In the second half of the 3rd century BC, the Boii allied with the other Cisalpine Gauls and the Etruscans against Rome. They also fought alongside Hannibal, killing the Roman general Lucius Postumius Albinus in 216 BC, whose skull was then turned into a sacrificial bowl.
A short time earlier, they had been defeated at the Battle of Telamon in 225 BC, and were again at Placentia in 194 BC (modern Piacenza) and Mutina in 193 BC (modern Modena). Publius Cornelius Scipio Nasica completed the Roman conquest of the Boii in 191 BC, celebrating a triumph for it. After their losses, according to Strabo, a large portion of the Boii left Italy.
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Boii on the Danube.
Contrary to the interpretation of the classical writers, the Pannonian Boii attested in later sources are not simply the remnants of those who had fled from Italy, but rather another division of the tribe, which had settled there much earlier.
The burial rites of the Italian Boii show many similarities with contemporary Bohemia, such as inhumation, which was uncommon with the other Cisalpine Gauls, or the absence of the typically western Celtic torcs.
This makes it much more likely that the Cisalpine Boii had actually originated from Bohemia rather than the other way round.
Having migrated to Italy from north of the Alps, some of the defeated Celts simply moved back to their kinsfolk.
The Pannonian Boii are mentioned again in the late 2nd century BC when they repelled the Cimbri and Teutones (Strabo VII, 2, 2). Later on, they attacked the city of Noreia (in modern Austria) shortly before a group of Boii (32,000 according to Julius Caesar) joined the Helvetii in their attempt to settle in western Gaul.
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After the Helvetian defeat at Bibracte, the influential Aedui tribe allowed the Boii survivors to settle on their territory, where they occupied the "oppidum" of Gorgobina. Although attacked by Vercingetorix during one phase of the war, they supported him with two thousand troops at the battle of Alesia (Caesar, "Commentarii de Bello Gallico", VII, 75).
Again, other parts of the Boii had remained closer to their traditional home, and settled in the Slovak and Hungarian lowlands by the Danube and the Mura, with a centre at Bratislava.
Dacian Conquest.
In the middle of the 1st century BC, the Boii tried to expand eastwards into modern-day Hungary, but clashed with the rising power of the Dacians under their king Burebista and were defeated. This war is often dated to the 60s or 50s BC or even precisely to 60/59 BC, but cannot be dated with that certainty. The numismatic material suggests that the clash may in fact have only happened by 41/40 BC. The Dacians under Burebista likely used a combination of military force and political strategies to conquer the Boii and compel some of them to migrate.
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Once the Boii were defeated or weakened, the Dacians would have annexed their territory, incorporating it into their expanding kingdom. If the early dating of the clash with Burebista is accepted, the migration of the Boii to Gaul and other parts of Europe may have been a consequence of their defeat and the Dacian occupation of their lands, as they sought new territories and opportunities elsewhere. However, specific details of this conquest and migration are often scarce in historical records, leaving much open to interpretation.
When the Romans finally conquered Pannonia in 8 AD, the Boii seem not to have opposed them. Their former territory was now called "deserta Boiorum" (deserta meaning 'empty or sparsely populated lands').
However, the Boii had not been exterminated: There was a "civitas Boiorum et Azaliorum" (the Azalii being a neighbouring tribe) which was under the jurisdiction of a prefect of the Danube shore ("praefectus ripae Danuvii"). This , a common Roman administrative term designating both a city and the tribal district around it, was later adjoined to the city of Carnuntum.
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The Boii in ancient sources.
Plautus.
Plautus refers to the Boii in "Captivi":
There is a play on words: "Boia" means 'woman of the Boii', also 'convicted criminal's restraint collar'.
Livy.
In volume 21 of his "History of Rome", Livy (59 BC – 17 AD) claims that it was a Boio man that offered to show Hannibal the way across the Alps.
Inscriptions.
In the first century BC, the Boii living in an oppidum of Bratislava minted Biatecs, high-quality coins with inscriptions (probably the names of kings) in Latin letters. At the oppidum of Manching there was a ceramic found bearing the labeling "Boius" or "Baius" which is being displayed at the local Celts and Romans museum. |
Backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back at least 1,600 years. The earliest record of backgammon itself dates to 17th-century England, being descended from the 16th-century game of Irish.
Backgammon is a two-player game of contrary movement in which each player has fifteen pieces known traditionally as men (short for "tablemen"), but increasingly known as "checkers" in the United States in recent decades. The backgammon table pieces move along twenty-four "points" according to the roll of two dice. The objective of the game is to move the fifteen pieces around the board and be first to "bear off", i.e., remove them from the board. The achievement of this while the opponent is still a long way behind results in a triple win known as a "backgammon", hence the name of the game.
Backgammon involves a combination of strategy and luck from rolling dice. While the dice may determine the outcome of a single game, the better player will accumulate the better record over a series of many games. With each roll of the dice, players must choose from numerous options for moving their pieces and anticipate possible counter-moves by the opponent. The optional use of a doubling cube allows players to raise the stakes during the game.
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History.
The earliest specific reference to backgammon was in a letter dated 1635, when it was emerging as a variant of the popular medieval Anglo-Scottish game of Irish; the latter was described as a better game. By the 19th century, however, backgammon had spread to Europe, where it rapidly superseded other tables games like Trictrac in popularity, and also to America, where the doubling cube was introduced. In other parts of the world, different tables games such as Nard or Nardy are better known.
Tables games.
Backgammon is a recent member of the large family of tables games that date back to ancient times. Its equipment is similar or identical to earlier tables games that have been depicted for centuries in art, leading to the mistaken belief that backgammon itself is much older.
Ancient history.
The history of board games can be traced back nearly 5,000 years to archaeological discoveries of the Jiroft culture, located in present-day Iran, the world's oldest game set having been discovered in the region with equipment comprising a dumbbell-shaped board, counters and dice. Although its precise rules are unknown, it has been termed the Game of 20 Squares and Irving Finkel has suggested a possible reconstruction. The Royal Game of Ur from 2600 BC may also be an ancestor or intermediate of modern-day table games like backgammon and is the oldest game for which rules have been handed down. It used tetrahedral dice. Various other board games spanning the 10th to 7th centuries BC have been found throughout modern day Iraq, Syria, Egypt and western Iran.
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Sasanian Empire.
The Persian tables game of nard or nardšir emerged somewhere between the 3rd and 6th century AD, one text ("Kār-nāmag ī Ardaxšēr ī Pāpakān") linking it with Ardashir I (r. 224–41), founder of the Sasanian dynasty, whereas another ("Wičārišn ī čatrang ud nihišn ī nēw-ardaxšēr") attributes it to Bozorgmehr Bokhtagan, the Vizier of Khosrow (r. 531–79), who is credited with the invention of the game.
Roman and Byzantine Empires.
The earliest identifiable tables game, Tabula, meaning 'table' or 'board', is described in an epigram of Byzantine Emperor Zeno (AD 476–491). The overall aim was to be first to bear one's pieces off; the board had the typical tables layout, with 24 points, 12 on each side; and there were 15 counters per player. However, unlike modern Western backgammon, there were three cubical dice not two, no bar nor doubling die, and all counters started off the board. Modern backgammon follows the same rules as tabula for hitting a blot and for bearing off; and the rules for re-entering pieces in backgammon are the same as those for initially entering pieces in tabula. The name Tavli () is still used in Greece for various tables games, which are frequently played in town plateias and cafes.
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The of Emperor Zeno's time is believed to be a direct descendant of the earlier Roman "ludus duodecim scriptorum" ('Game of twelve lines') with the board's middle row of points removed, and only the two outer rows remaining. used a board with three rows of 12 points each, with the 15 pieces being moved in opposing directions by the two players across three rows according to the roll of the three cubical dice. Little specific text about the gameplay of has survived; it may have been related to the older Ancient Greek dice game "Kubeia". The earliest known mention of the game is in Ovid's "Ars Amatoria" ('The Art of Love'), written between 1 BC and 8 AD. In Roman times, this game was also known as "alea".
Western Europe.
Tables games first appeared in France during the 11th century and became a favourite pastime of gamblers. In 1254, Louis IX issued a decree prohibiting his court officials and subjects from playing. They were played in Germany in the 12th century, and had reached Iceland by the 13th century. In Spain, the Alfonso X manuscript "Libro de los Juegos", completed in 1283, describes rules for a number of dice and table games in addition to its discussion of chess. By the 17th century, games at tables had spread to Sweden. A wooden board and counters were recovered from the wreck of the "Vasa" among the belongings of the ship's officers. Tables games appear widely in paintings of this period, mainly those of Dutch and German painters, such as van Ostade, Jan Steen, Hieronymus Bosch, and Bruegel. Among surviving artworks are "Cardsharps" by Caravaggio.
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Backgammon.
Early backgammon.
Backgammon's immediate predecessor was the 16th century tables game of Irish. Irish was the Anglo-Scottish equivalent of the French "Toutes Tables" and Spanish "Todas Tablas", the latter name first being used in the 1283 "El Libro de los Juegos", a translation of Arabic manuscripts by the Toledo School of Translators. Irish had been popular at the Scottish court of James IV and considered to be "the more serious and solid game" when the variant which became known as Backgammon began to emerge in the first half of the 17th century. In medieval Italy, Barail was played on a backgammon board, with the important difference that both players moved their pieces counter-clockwise and starting from the same side of the board. The game rules for Barail are recorded in a 13th-century manuscript held in the Italian National Library in Florence.
The earliest mention of backgammon, under the name "Baggammon", was by James Howell in a letter dated 1635. In English, the word "backgammon" is most likely derived from "back" and , meaning "game" or "play". Meanwhile, the first use documented by the Oxford English Dictionary was in 1650. In 1666, it is reported that the "old name for backgammon used by Shakespeare and others" was Tables. However, it is clear from Willughby that "tables" was a generic name and that the phrase "playing at tables" was used in a similar way to "playing at cards". The first known rules of "Back Gammon" were produced by Francis Willoughby around 1672; they were quickly followed by Charles Cotton in 1674.
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In the 16th century, Elizabethan laws and church regulations had prohibited "playing at tables" in England, but by the 18th century, Backgammon had superseded Irish and become popular among the English clergy. Edmond Hoyle published "A Short Treatise on the Game of Back-Gammon" in 1753; this described rules and strategy for the game and was bound together with a similar text on whist.
The early form of backgammon was very similar to its predecessor, Irish. The aim, board, number of pieces or "men", direction of play and starting layout were the same as in the modern game. However, there was no doubling die, there was no bar on the board or the bar was not used (men simply being moved off the table when hit) and the scoring was different. The game was won double if either the winning throw was a doublet or the opponent still had men outside the home board. It was won triple if a player bore all men off before any of the opponent's men reached the home board; this was a "back-gammon". Some terminology, such as "point", "hitting a blot", "home", "doublet", "bear off" and "men" are recognisably the same as in the modern game; others, such as "binding a man" (adding a second man to a point) "binding up the tables" (taking all one's first 6 points), "fore game", "latter game", "nipping a man" (hitting a blot and playing it on forwards) "playing at length" (using both dice to move one man) are no longer in vogue.
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Modern backgammon.
By no later than 1850, the rules of play had changed to those used today. Tables boards were now made with a "bar" in the centre and men that were hit went onto the bar. Winning double or by "two hits" was achieved by bearing all one's men off before the other has borne this was now called a "gammon". If the winner bore off all men while the loser still had men in his adversary's table, it was a "back-gammon" and worth "three hits", i.e., triple.
The most recent major development in backgammon was the addition of the doubling cube. Doubles had originally been recorded by placing "common parlour matches" on the bar in the centre of the board. A doubling cube was first introduced in the 1920s in New York City among members of gaming clubs in the Lower East Side. The cube required players not only to select the best move in a given position, but also to estimate the probability of winning from that position, transforming backgammon into the expected value-driven game played in the 20th and 21st centuries.
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The popularity of backgammon surged in the mid-1960s, in part due to the charisma of Prince Alexis Obolensky who became known as "The Father of Modern Backgammon". "Obe", as he was called by friends, co-founded the International Backgammon Association, which published a set of official rules. He also established the World Backgammon Club of Manhattan, devised a backgammon tournament system in 1963, then organized the first major international backgammon tournament in March 1964, which attracted royalty, celebrities and the press. The game became a huge fad and was played on college campuses, in discothèques and at country clubs; stockbrokers and bankers began playing at conservative men's clubs. People young and old all across the country dusted off their boards and pieces. Cigarette, liquor and car companies began to sponsor tournaments, and Hugh Hefner held backgammon parties at the Playboy Mansion. Backgammon clubs were formed and tournaments were held, resulting in a World Championship promoted in Las Vegas in 1967.
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In the second half of the 20th century, new terms were introduced in America, such as 'beaver' and 'checkers' for men (although American backgammon experts Jacoby and Crawford continued to use both the older terms as well as the new ones).
Most recently, the United States Backgammon Federation (USBGF) was organized in 2009 to repopularize the game in the United States. Board and committee members include many of the top players, tournament directors and writers in the worldwide backgammon community. The USBGF has recently created Standards of Ethical Practice to address issues on which tournament rules fail to touch.
In its country of origin, the UK Backgammon Federation is the national authority and runs a backgammon the Backgammon Galaxy UK Open as well as club championships, online leagues and knockout tournaments. Like the USBGF they are active members of the World Backgammon Federation (WBF) and their tournament rules have been adopted in their entirety by the WBF.
Software.
Backgammon entered the computer era in the 1990s when software was developed to play and analyze games, and for people to play one another over the internet.
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Real-time online play began with the First Internet Backgammon Server in July 1992, but there are now a range of options.
Rules.
Since 2018, backgammon has been overseen internationally by the World Backgammon Federation who set the rules of play for international tournaments.
Backgammon playing pieces may be termed men, checkers, draughts, stones, counters, pawns, discs, pips, chips, or nips. Checkers is a relatively modern American English term derived from another board game, draughts, which in US English is called checkers.
The objective is for players to bear off all their disc pieces from the board before their opponent can do the same. As the playing time for each individual game is short, it is often played in matches where victory is awarded to the first player to reach a certain number of points.
Board.
The dimensions of a board when opened, for a tournament game, should be at a minimum of 44 cm by 55 cm to a maximum of 66 cm by 88 cm.
Setup.
Each side of the board has a track of 12 isosceles triangles, called points. The points form a continuous track in the shape of a horseshoe, and are numbered from 1 to 24. In the most commonly used setup, each player begins with fifteen pieces; two are placed on their 24-point, three on their 8-point, and five each on their 13-point and their 6-point. The two players move their pieces in opposing directions, from the 24-point towards the 1-point.
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Points 1 through 6 are called the home board or inner board, and points 7 through 12 are called the outer board. The 7-point is referred to as the bar point, and the 13-point as the midpoint. The 5-point for each player is sometimes called the "golden point".
Movement.
To start the game, each player rolls one die, and the player with the higher number moves first using the numbers shown on both dice. If the players roll the same number, they must roll again until they roll different numbers. Both dice must land completely flat on the right-hand side of the gameboard. The players then take alternate turns, rolling two dice at the beginning of each turn.
After rolling the dice, players must, if possible, move their pieces according to the number shown on each die. For example, if the player rolls a 6 and a 3 (denoted as "6-3"), the player must move one checker six points forward, and another or the same checker three points forward. The same checker may be moved twice, as long as the two moves can be made separately and legally: six and then three, or three and then six. If a player rolls two of the same number, called doubles, that player must play each die twice. For example, a roll of 5-5 allows the player to make four moves of five spaces each. On any roll, a player must move according to the numbers on both dice if it is at all possible to do so. If one or both numbers do not allow a legal move, the player forfeits that portion of the roll and the turn ends. If moves can be made according to either one die or the other, but not both, the higher number must be used. If one die is unable to be moved, but such a move is made possible by the moving of the other die, that move is compulsory.
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In the course of a move, a checker may land on any point that is unoccupied or is occupied by one or more of the player's own checkers. It may also land on a point occupied by exactly one opposing checker, or "blot". In this case, the blot has been "hit" and is placed in the middle of the board on the bar that divides the two sides of the playing surface. A checker may never land on a point occupied by two or more opposing checkers; thus, no point is ever occupied by checkers from both players simultaneously. There is no limit to the number of checkers that can occupy a point or the bar at any given time.
Checkers placed on the bar must re-enter the game through the opponent's home board before any other move can be made. A roll of 1 allows the checker to enter on the 24-point (opponent's 1), a roll of 2 on the 23-point (opponent's 2), and so forth, up to a roll of 6 allowing entry on the 19-point (opponent's 6). Checkers may not enter on a point occupied by two or more opposing checkers. Checkers can enter on unoccupied points, or on points occupied by a single opposing checker; in the latter case, the single checker is hit and placed on the bar. A player may not move any other checkers until all checkers belonging to that player on the bar have re-entered the board. If a player has checkers on the bar, but rolls a combination that does not allow any of those checkers to re-enter, the player does not move. If the opponent's home board is completely "closed" (i.e. all six points are each occupied by two or more checkers), there is no roll that will allow a player to enter a checker from the bar, and that player stops rolling and playing until at least one point becomes open (occupied by one or zero checkers) due to the opponent's moves.
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A turn ends only when the player has removed his or her dice from the board. Prior to this moment, a move can be undone and replayed an unlimited number of times.
Bearing off.
When all of a player's checkers are in that player's home board, that player may start removing them; this is called "bearing off". A roll of 1 may be used to bear off a checker from the 1-point, a 2 from the 2-point, and so on. If all of a player's checkers are on points lower than the number showing on a particular die, the player must use that die to bear off one checker from the highest occupied point. For example, if a player rolls a 6 and a 5, but has no checkers on the 6-point and two on the 5-point, then the 6 and the 5 must be used to bear off the two checkers from the 5-point. When bearing off, a player may also move a lower die roll before the higher even if that means the full value of the higher die is not fully utilized. For example, if a player has exactly one checker remaining on the 6-point, and rolls a 6 and a 1, the player may move the 6-point checker one place to the 5-point with the lower die roll of 1, and then bear that checker off the 5-point using the die roll of 6; this is sometimes useful tactically. As before, if there is a way to use all moves showing on the dice by moving checkers within the home board or by bearing them off, the player must do so. If a player's checker is hit while in the process of bearing off, that player may not bear off any others until it has been re-entered into the game and moved into the player's home board, according to the normal movement rules.
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The first player to bear off all fifteen of their own checkers wins the game. When keeping score in backgammon, the points awarded depend on the scale of the victory. A player who bears off all fifteen pieces when the opponent has borne off at least one, wins a "single game" worth 1 point. If all fifteen have been borne off before the opponent gets at least one checker off, this is a "gammon" or "double game" worth 2 points. A "backgammon" or "triple game" is worth 3 points and occurs when the losing player has borne off no pieces and has one or more on the bar and/or in the winner's home table (inner board).
Doubling cube.
To speed up match play and to provide an added dimension for strategy, a doubling cube is usually used. The doubling cube is not a die to be rolled, but rather a marker, with the numbers 2, 4, 8, 16, 32, and 64 inscribed on its sides to denote the current stake. At the start of each game, the doubling cube is placed on the midpoint of the bar with the number 64 showing; the cube is then said to be "centered, on 1". When the cube is still centered, either player may start their turn by proposing that the game be played for twice the current stakes. Their opponent must either accept ("take") the doubled stakes or resign ("drop") the game immediately.
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Whenever a player accepts doubled stakes, the cube is placed on their side of the board with the corresponding power of two facing upward, to indicate that the right to redouble, which is to offer to continue doubling the stakes, belongs exclusively to that player. If the opponent drops the doubled stakes, they lose the game at the current value of the doubling cube. For instance, if the cube showed the number 2 and a player wanted to redouble the stakes to put it at 4, the opponent choosing to drop the redouble would lose two, or twice the original stake.
There is no limit on the number of redoubles. Although 64 is the highest number depicted on the doubling cube, the stakes may rise to 128, 256, and so on. In money games, a player is often permitted to "beaver" when offered the cube, doubling the value of the game again, while retaining possession of the cube.
A variant of the doubling cube "beaver" is the "raccoon". Players who doubled their opponent, seeing the opponent beaver the cube, may in turn then double the stakes once again ("raccoon") as part of that cube phase before any dice are rolled. The opponent retains the doubling cube. An example of a "raccoon" is the following: White doubles Black to 2 points, Black accepts then beavers the cube to 4 points; White, confident of a win, raccoons the cube to 8 points, while Black retains the cube. Such a move adds greatly to the risk of having to face the doubling cube coming back at 8 times its original value when first doubling the opponent (offered at 2 points, counter offered at 16 points) should the luck of the dice change.
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Some players may opt to invoke the "Murphy rule" or the "automatic double rule". If both opponents roll the same opening number, the doubling cube is incremented on each occasion yet remains in the middle of the board, available to either player. The Murphy rule may be invoked with a maximum number of automatic doubles allowed and that limit is agreed to prior to a game or match commencing. When a player decides to double the opponent, the value is then a double of whatever face value is shown (e.g. if two automatic doubles have occurred putting the cube up to 4, the first in-game double will be for 8 points). The Murphy rule is not an official rule in backgammon and is rarely, if ever, seen in use at officially sanctioned tournaments.
The "Jacoby rule", named after Oswald Jacoby, allows gammons and backgammons to count for their respective double and triple values only if the cube has already been offered and accepted. This encourages a player with a large lead to double, possibly ending the game, rather than to play it to conclusion hoping for a gammon or backgammon. The Jacoby rule is widely used in money play but is not used in match play.
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The "Crawford rule", named after John R. Crawford, is designed to make match play more equitable for the player in the lead. If a player is one point away from winning a match, that player's opponent will always want to double as early as possible in order to catch up. Whether the game is worth one point or two, the trailing player must win to continue the match. To balance the situation, the Crawford rule requires that when a player first reaches a score one point short of winning, neither player may use the doubling cube for the following game, called the "Crawford game". After the Crawford game, normal use of the doubling cube resumes. The Crawford rule is routinely used in tournament match play. It is possible for a Crawford game to never occur in a match.
If the Crawford rule is in effect, then another option is the "Holland rule", named after Tim Holland, which stipulates that after the Crawford game, a player cannot double until after at least two rolls have been played by each side. It was common in tournament play in the 1980s, but is now rarely used.
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Related games.
Minor variations to the standard game are common among casual players in certain regions. For instance, only allowing a maximum of five men on any point (Britain) or disallowing "hit-and-run" in the home board (Middle East).
There are also many relatives of backgammon within the tables family with different aims, modes of play and strategies. Some are played primarily throughout one geographic region, and others add new tactical elements to the game. These other tables games commonly have a different starting position, restrict certain moves, or assign special value to certain dice rolls, but in some geographic games even the rules and direction of movement of the counters change, rendering them fundamentally different.
"Acey-deucey" is a relative of backgammon in which players start with no counters on the board, and must enter them onto the board at the beginning of the game. The roll of 1-2 is given special consideration, allowing the player, after moving the 1 and the 2, to select any desired doubles move. A player also receives an extra turn after a roll of 1-2 or of doubles.
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"Hypergammon" is a game in which players have only three counters on the board, starting with one each on the 24, 23 and 22 points. With the aid of a computer this game was solved by Hugh Sconyers around 1994, meaning that exact equities for all cube positions are available for all 32 million possible positions.
"Nard" is a traditional tables game from Persia which may be an ancestor of backgammon. It has a different opening layout in which all 15 pieces start on the 24th point. During play pieces may not be hit and there are no gammons or backgammons.
"Ban-sugoroku" is a Japanese game that is a close relative of backgammon. It utilizes the same starting position but has slightly different rules.
"Russian backgammon" is a variant described in 1895 as: "much in vogue in Russia, Germany, and other parts of the Continent". Players start with no counters on the board, and both players move in the same direction to bear off in a common home board. In this variant, doubles are powerful: four moves are played as in backgammon, followed by four moves according to the difference of the dice value from 7, and then the player has another turn (with the caveat that the turn ends if any portion of it cannot be completed).
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"Gul bara" and "Tapa" are tables games popular in south-eastern Europe and Turkey. The play will iterate among Backgammon, Gul Bara, and Tapa until one of the players reaches a score of 7 or 5.
"Coan ki" is an ancient Chinese tables game.
"Plakoto", "Fevga" and "Portes" are three varieties of tables games played in Greece. Together, the three are referred to as "Tavli" and are usually played one after the other; game being three, five, or seven points.
"Misere (backgammon to lose)" is a variant of backgammon in which the objective is to lose the game.
"Tavla" is a Turkish variation.
Strategy and tactics.
Backgammon is played in two principal variations, "money" and "match" play:
The format has a significant effect on strategy. In a match, the objective is not to win the maximum possible number of points, but rather to simply reach the score needed to win the match, so optimal play may depend on the match score. In money play, the theoretically correct checker play and cube action would never vary based on the score.
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Backgammon has an established opening theory, although it is less detailed than that of chess. The tree of positions expands rapidly because of the number of possible dice rolls and the moves available on each turn. Recent computer analysis has offered more insight on opening plays, but the midgame is reached quickly. After the opening, backgammon players frequently rely on some established general strategies, combining and switching among them to adapt to the changing conditions of a game.
There are several strategies or "game plans" to achieve a win:
A "backgame" is a strategy that involves holding two or more anchors in an opponent's home board while being substantially behind in the race. The anchors obstruct the opponent's checkers and create opportunities to hit them as they move home. The backgame is generally used only to salvage a game wherein a player is already significantly behind. Using a backgame as an initial strategy is usually unsuccessful.
"Duplication" refers to the placement of checkers such that one's opponent needs the same dice rolls to achieve different goals. For example, players may position all of their blots in such a way that the opponent must roll a 2 in order to hit any of them, reducing the probability of being hit more than once.
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"Diversification" refers to a complementary tactic of placing one's own checkers in such a way that more numbers are useful.
The "pipcount" is number of pips needed to move a player's checkers around and off the board. Many positions require a measurement of a player's standing in the race, for example, in making a doubling cube decision, or in determining whether to run home and begin bearing off. The difference between the two players' pip counts is a measure of the leader's racing advantage. For cube decisions, a number of formulas have been developed over the years, including the Thorpe count, the Ward count, the Keith count, and iSight. These calculations enable a player to determine whether to offer or take a double based on the pipcount in non-contact positions.
Cube handling.
Two theoretical models provide a basis for cube handling, i.e. when to offer a double and when to accept an offered double. Both ignore the effects of gammons and backgammons.
In practice, the takepoints and doubling points are somewhere in between, since while cube ownership cannot be ignored, assuming maximal efficiency for a re-cube is also not a valid assumption. Ignoring gammons and backgammons, the takepoint in money play is about 22%. All of the above ignores gammons and backgammons for either side, so in practice the calculation of takepoints is more complicated.
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Equity.
A player's equity in a money or unlimited game is the average expected value that will be won or lost as a result of that game. For instance, if a player is certain to win but has no chance of a gammon or backgammon their equity is 1 and their opponent's equity is −1. If it is certain that the player will win a backgammon, their equity is 3 and their opponent's equity is −3. In Example 1 below, the player's winning chances are 75%, which corresponds to an equity of +0.5.
Cheating.
To reduce the possibility of cheating, most good-quality backgammon sets use precision dice and a dice cup. This reduces the likelihood of loaded dice being used, which is the main way of cheating in face-to-face play. A common method of cheating online is the use of a computer program to find the optimal move on each turn; to combat this, many online sites use move-comparison software that identifies when a player's moves resemble those of a backgammon program. Online cheating has therefore become extremely difficult.
Social and competitive play.
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Legality.
In "State of Oregon v. Barr", a 1982 court case pivotal to the continued widespread organised playing of backgammon in the US, the State argued that backgammon is a game of chance and that it was therefore subject to Oregon's stringent gambling laws. Paul Magriel was a key witness for the defence, contradicting Roger Nelson, the expert prosecution witness, by saying, "Game theory, however, really applies to games with imperfect knowledge, where something is concealed, such as poker. Backgammon is not such a game. Everything is in front of you. The person who uses that information in the most effective manner will win." After the closing arguments, Judge Stephen S. Walker concluded that backgammon is a game of skill, not a game of chance, and found the defendant, backgammon tournament director Ted Barr, not guilty of promoting gambling.
Club and tournament play.
Played "ad hoc" in cafés and bars, clubs throughout Europe also host backgammon with informal gatherings to play throughout the day or in the evening as well as by way of social interaction. A few clubs offer specialized backgammon services, maintaining their own facilities or offering computer analysis of troublesome plays. Around 2003, some club leaders noticed a growth of interest in backgammon, and attributed it to the game's popularity on the internet.
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