task stringlengths 0 154k | __index_level_0__ int64 0 39.2k |
|---|---|
A round clock only has three hands: hour, minute, second. All hands look identical and move continuously. Moreover, there is no number or reference mark so that the "upright position" is unknown. The clock functions the same as a normal 12-hour analogue clock. Despite the inconvenient design, for most time it is possib... | 25,900 |
Consider a circle where 2n distinct points have been marked on its circumference. A cutting C consists of connecting the 2n points with n line segments, so that no two line segments intersect, including on their end points. The n line segments then cut the circle into n + 1 pieces.
Each piece is painted either black or... | 25,901 |
Define M(n) to be the minimum number of matchsticks needed to represent the number n. A number can be represented in digit form or as an expression involving addition and/or multiplication. Also order of operations must be followed, that is multiplication binding tighter than addition. Any other symbols or operations, ... | 25,902 |
Consider a unit circle circle with radius 1 C_0 on the plane that does not enclose the origin. For k\ge 1, a circle C_k is created by scaling and rotating C_{k - 1} with respect to the origin . That is, both the radius and the distance to the origin are scaled by the same factor, and the centre of rotation is the origi... | 25,903 |
Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to make a ... | 25,904 |
A contiguous range of positive integers is called a divisible range if all the integers in the range can be arranged in a row such that the n-th term is a multiple of n. For example, the range [6..9] is a divisible range because we can arrange the numbers as 7,6,9,8. In fact, it is the 4th divisible range of length 4, ... | 25,905 |
Let G(n) denote the largest possible area of an n-gon a polygon with n sides contained in the region \{(x, y) \in \Bbb R^2: x^4 \leq y \leq 1\}. For example, G(3) = 1 and G(5)\approx 1.477309771. Find G(101) rounded to nine digits after the decimal point. | 25,906 |
Claire Voyant is a teacher playing a game with a class of students.
A fair coin is tossed on the table. All the students can see the outcome of the toss, but Claire cannot.
Each student then tells Claire whether the outcome is head or tail. The students may lie, but Claire knows the probability that each individual stu... | 25,907 |
Two players play a game with two piles of stones. The players alternately take stones from one or both piles, subject to: the total number of stones taken is equal to the size of the smallest pile before the move; the move cannot take all the stones from a pile. The player that is unable to move loses. For example, if ... | 25,908 |
Two players play a game with at least two piles of stones. The players alternately take stones from one or more piles, subject to: the total number of stones taken is equal to the size of the smallest pile before the move; the move cannot take all the stones from a pile. The player that is unable to move loses. For exa... | 25,909 |
A driller drills for water. At each iteration the driller chooses a depth d (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby. Drilling to depth d tak... | 25,910 |
A permutation \pi of \{1, \dots, n\} can be represented in one-line notation as \pi(1),\ldots,\pi(n) . If all n! permutations are written in lexicographic order then \textrm{rank}(\pi) is the position of \pi in this 1-based list. For example, \text{rank}(2,1,3) = 3 because the six permutations of \{1, 2, 3\} in lexicog... | 25,911 |
A permutation \pi of \{1, \dots, n\} can be represented in one-line notation as \pi(1),\ldots,\pi(n) . If all n! permutations are written in lexicographic order then \textrm{rank}(\pi) is the position of \pi in this 1-based list. For example, \text{rank}(2,1,3) = 3 because the six permutations of \{1, 2, 3\} in lexicog... | 25,912 |
Given a right-angled triangle with integer sides, the smaller angle formed by the two medians drawn on the the two perpendicular sides is denoted by \theta. Let f(\alpha, L) denote the sum of the sides of the right-angled triangle minimizing the absolute difference between \theta and \alpha among all right-angled trian... | 25,913 |
Three epistemologists, known as A, B, and C, are in a room, each wearing a hat with a number on it. They have been informed beforehand that all three numbers are positive and that one of the numbers is the sum of the other two. Once in the room, they can see the numbers on each other's hats but not on their own. Starti... | 25,914 |
Three friends attempt to collectively choose one of n options, labeled 1,\dots,n, based upon their individual preferences. They choose option i if for every alternative option j at least two of the three friends prefer i over j. If no such option i exists they fail to reach an agreement. Define P(n) to be the probabili... | 25,915 |
An infant's toy consists of n cups, labelled C_1,\dots,C_n in increasing order of size. The cups may be stacked in various combinations and orientations to form towers. The cups are shaped such that the following means of stacking are possible: Nesting: C_k may sit snugly inside C_{k+1}. Base-to-base: C_{k+2} or C_{k-2... | 25,916 |
A clock sequence is a periodic sequence of positive integers that can be broken into contiguous segments such that the sum of the n-th segment is equal to n. For example, the sequence 1\ 2\ 3\ 4\ 3\ 2\ 1\ 2\ 3\ 4\ 3\ 2\ 1\ 2\ 3\ 4\ 3\ 2\ 1\ \cdots is a clock sequence with period 6, as it can be broken into 1\Big |2\Big... | 25,917 |
An L-expression is defined as any one of the following: a natural number; the symbol A; the symbol Z; the symbol S; a pair of L-expressions u, v, which is written as u(v). An L-expression can be transformed according to the following rules: A(x) \to x + 1 for any natural number x; Z(u)(v) \to v for any L-expressions u,... | 25,918 |
An L-expression is defined as any one of the following: a natural number; the symbol A; the symbol Z; the symbol S; a pair of L-expressions u, v, which is written as u(v). An L-expression can be transformed according to the following rules: A(x) \to x + 1 for any natural number x; Z(u)(v) \to v for any L-expressions u,... | 25,919 |
An irrational number x can be uniquely expressed as a continued fraction [a_0; a_1,a_2,a_3,\dots]:
x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}
where a_0 is an integer and a_1,a_2,a_3,\dots are positive integers. Define k_j(x) to be the geometric mean of a_1,a_2,\dots,a_j. That is, k_j(x)=(a_1a_2 \cd... | 25,920 |
Let s_n be the n-th positive integer that does not contain three consecutive ones in its binary representation. For example, s_1 = 1 and s_7 = 8. Define F(N) to be the sum of n^2 for all n\leq N where s_n is odd. You are given F(10)=199. Find F(10^{16}) giving your answer modulo 10^9+7. | 25,921 |
The numbers from 1 to 12 can be arranged into a 3 \times 4 matrix in either row-major or column-major order:
R=\begin{pmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\end{pmatrix}, C=\begin{pmatrix}
1 & 4 & 7 & 10\\
2 & 5 & 8 & 11\\
3 & 6 & 9 & 12\end{pmatrix}
By swapping two entries at a time, at least 8 swap... | 25,922 |
For a given integer R consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius R. Define F(R) to be the largest inradius of those triangles. You are given F(100) = 36. Find F(10^{18}). | 25,923 |
The function s(n) is defined recursively for positive integers by
s(1) = 1 and s(n+1) = \big(s(n) - 1\big)^3 +2 for n\geq 1. The sequence begins: s(1) = 1, s(2) = 2, s(3) = 3, s(4) = 10, \ldots. For positive integers N, define T(N) = \sum_{a=1}^N \sum_{b=1}^N \gcd\Big(s\big(s(a)\big), s\big(s(b)\big)\Big). You are gi... | 25,924 |
Let P(n) be the number of permutations of \{1,2,3,\ldots,2n\} such that: 1. There is no ascending subsequence with more than n+1 elements, and 2. There is no descending subsequence with more than two elements. Note that subsequences need not be contiguous. For example, the permutation (4,1,3,2) is not counted because i... | 25,925 |
The sequence s_n is defined by s_1 = 102022661 and s_n = s_{n-1}^2 \bmod {998388889} for n > 1. Let a_n = s_{2n - 1} and b_n = s_{2n} for n=1,2,... Define an N \times N matrix whose values are M_{i,j} = a_i + b_j. Let A(N) be the minimal path sum from M_{1,1} (top left) to M_{N,N} (bottom right), where each step is eit... | 25,926 |
The sequence a_n is defined by a_1=1, and then recursively for n\geq1:
\begin{align*}
a_{2n} &=2a_n\\
a_{2n+1} &=a_n-3a_{n+1}
\end{align*}
The first ten terms are 1, 2, -5, 4, 17, -10, -17, 8, -47, 34. Define \displaystyle S(N) = \sum_{n=1}^N a_n. You are given S(10) = -13 Find S(10^{12}). | 25,927 |
We call a triangle fortunate if it has integral sides and at least one of its vertices has the property that the distance from it to the triangle's orthocentre is exactly half the distance from the same vertex to the triangle's circumcentre . Triangle ABC above is an example of a fortunate triangle with sides (6,7,8). ... | 25,928 |
For a positive integer n we define \tau(n) to be the count of the divisors of n. For example, the divisors of 12 are \{1,2,3,4,6,12\} and so \tau(12) = 6. A positive integer n is a tau number if it is divisible by \tau(n). For example \tau(12)=6 and 6 divides 12 so 12 is a tau number. Let m(k) be the smallest tau numbe... | 25,929 |
Consider the following recurrence relation:
\begin{align}
a_0 &= \frac{\sqrt 5 + 1}2\\
a_{n+1} &= \dfrac{a_n(a_n^4 + 10a_n^2 + 5)}{5a_n^4 + 10a_n^2 + 1}
\end{align} Note that a_0 is the golden ratio . a_n can always be written in the form \dfrac{p_n\sqrt{5}+1}{q_n}, where p_n and q_n are positive integers. Let s(n)=p_n... | 25,930 |
A Young diagram is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such that the left-most squares of all rows are aligned vertically; the top squares of all columns are aligned horizontally; the rows are non-increasing in size as we move top to bottom; the columns are non... | 25,931 |
Counting DNA Nucleotides
Description:
A Rapid Introduction to Molecular Biology
Figure 1
.
A 1900 drawing by Edmund Wilson of onion cells at different stages of mitosis. The sample has been dyed, causing chromatin in the cells (which soaks up the dye) to appear in greater contrast to the rest of the cell.
Figur... | 25,932 |
Transcribing DNA into RNA
Description:
The Second Nucleic Acid
Figure 1
.
Structural differences between RNA and DNA
In
“Counting DNA Nucleotides”
, we described the
primary structure
of a
nucleic acid
as a
polymer
of
nucleotide
units,
and we mentioned that the omnipresent nucleic acid
DNA
is compose... | 25,933 |
Complementing a Strand of DNA
Description:
The Secondary and Tertiary Structures of DNA
Figure 1
.
Base pairing across the two strands of DNA.
Figure 2
.
The double helix of DNA on the molecular scale.
In
“Counting DNA Nucleotides”
, we introduced
nucleic acids
, and we saw that the
primary structure
o... | 25,934 |
Rabbits and Recurrence Relations
Description:
Wascally Wabbits
Figure 1
.
The growth of Fibonacci's rabbit population for the first six months.
Figure 2
.
Erosion at Lake Mungo in New South Wales, which was initiated by European rabbits in the 19th Century. Courtesy Pierre Pouliquin.
In 1202, Leonardo of P... | 25,935 |
Computing GC Content
Description:
Identifying Unknown DNA Quickly
Figure 1
.
The table above was computed from a large number of English words and shows for any letter the frequency with which it appears in those words. These frequencies can be used to reliably identify a piece of English text and differentiate it ... | 25,936 |
Counting Point Mutations
Description:
Evolution as a Sequence of Mistakes
Figure 1
.
A point mutation in DNA changing a C-G pair to an A-T pair.
A
mutation
is simply a mistake that occurs during the creation or copying of a
nucleic acid
, in particular
DNA
. Because nucleic acids are vital to
cellular
fu... | 25,937 |
Mendel's First Law
Description:
Introduction to Mendelian Inheritance
Figure 1
.
A Punnett square representing the possible outcomes of crossing a heterozygous organism (Yy) with a homozygous recessive organism (yy); here, the dominant allele Y corresponds to yellow pea pods, and the recessive allele y corresponds ... | 25,938 |
Translating RNA into Protein
Description:
The Genetic Code
Figure 1
.
The human hemoglobin molecule consists of 4 polypeptide chains; α subunits are shown in red and β subunits are shown in blue
Just as
nucleic acids
are
polymers
of
nucleotides
,
proteins
are chains of
smaller molecules called
amino aci... | 25,939 |
Finding a Motif in DNA
Description:
Combing Through the Haystack
Figure 1
.
The human chromosomes stained with a probe for Alu elements, shown in green.
Finding the same interval of DNA in the
genomes
of two different organisms
(often taken from different species) is highly suggestive that the interval has
th... | 25,940 |
Consensus and Profile
Description:
Finding a Most Likely Common Ancestor
In
“Counting Point Mutations”
, we calculated the minimum number of symbol mismatches between two strings of equal length
to model the problem of finding the minimum number of point mutations occurring
on the evolutionary path between two
hom... | 25,941 |
Mortal Fibonacci Rabbits
Description:
Wabbit Season
Figure 1
.
A c.1905 photo from Australia of a cart loaded to the hilt with rabbit skins.
Figure 2
.
Western Australia's rabbit fence is actually not the longest fence in the world as the sign claims. That honor goes to a 3,500 mile fence in southeastern Austr... | 25,942 |
Overlap Graphs
Description:
A Brief Introduction to Graph Theory
Networks arise everywhere in the practical world, especially in biology. Networks are
prevalent in popular applications such as modeling the spread of disease, but the extent
of network applications spreads far beyond popular science. Our first quest... | 25,943 |
Calculating Expected Offspring
Description:
The Need for Averages
Averages arise everywhere. In sports, we want to project the average number of games
that a team is expected to win; in gambling, we want to project the average losses
incurred playing blackjack; in business, companies want to calculate their average... | 25,944 |
Finding a Shared Motif
Description:
Searching Through the Haystack
In
“Finding a Motif in DNA”
, we searched a given
genetic string
for a
motif
; however, this problem assumed that we know the motif in advance. In practice,
biologists often do not know exactly what they are looking for. Rather, they must
hunt th... | 25,945 |
Independent Alleles
Description:
Mendel's Second Law
Figure 1
.
Mendel's second law dictates that every one of the 16 possible assignments of parental alleles is equally likely. The Punnett square for two factors therefore places each of these assignments in a cell of a 4 X 4 table. The probability of an offspring'... | 25,946 |
Finding a Protein Motif
Description:
Motif Implies Function
Figure 1
.
The human cyclophilin family, as represented by the structures of the isomerase domains of some of its members.
As mentioned in
“Translating RNA into Protein”
,
proteins
perform every practical function in the
cell
.
A structural and fun... | 25,947 |
Inferring mRNA from Protein
Description:
Pitfalls of Reversing Translation
When researchers discover a new
protein
, they would like to infer
the strand of
mRNA
from which this protein could have been
translated
,
thus allowing them to locate
genes
associated with this protein on the
genome
.
Unfortunately,... | 25,948 |
Open Reading Frames
Description:
Transcription May Begin Anywhere
Figure 1
.
Schematic image of the particular ORF with start and stop codons shown.
In
“Transcribing DNA into RNA”
, we discussed the
transcription
of
DNA
into
RNA
, and in
“Translating RNA into Protein”
, we examined
the
translation
of R... | 25,949 |
Enumerating Gene Orders
Description:
Rearrangements Power Large-Scale Genomic Changes
Figure 1
.
Similar regions in mouse and human chromosomes. Image credit: U.S. Department of Energy Human Genome Program
Point mutations
can create changes in populations of organisms from the same species,
but they lack the p... | 25,950 |
Calculating Protein Mass
Description:
Chaining the Amino Acids
Figure 1
.
Formation of a peptide bond
Figure 2
.
Outermost acids
In
“Translating RNA into Protein”
, we examined the
translation
of
RNA
into an
amino acid
chain for the construction
of a
protein
. When two amino acids link together, th... | 25,951 |
Locating Restriction Sites
Description:
The Billion-Year War
Figure 1
.
DNA cleaved by EcoRV restriction enzyme
The war between viruses and bacteria has been waged for over a billion years.
Viruses called
bacteriophages
(or simply phages) require
a bacterial host to propagate, and so they must somehow infiltr... | 25,952 |
RNA Splicing
Description:
Genes are Discontiguous
Figure 1
.
The elongation of a pre-mRNA by RNAP as it moves down the template strand of DNA.
Figure 2
.
RNA is identical to the coding strand except for the replacement of thymine with uracil.
In
“Transcribing DNA into RNA”
, we mentioned that a
strand
o... | 25,953 |
Enumerating k-mers Lexicographically
Description:
Organizing Strings
When cataloguing a collection of
genetic strings
, we should have an established system by which to organize
them. The standard method is to organize strings as they would appear in a dictionary, so that
"APPLE" precedes "APRON", which in turn co... | 25,954 |
Longest Increasing Subsequence
Description:
A Simple Measure of Gene Order Similarity
In
“Enumerating Gene Orders”
, we started talking about comparing the order of
genes
on a
chromosome
taken from two different species and moved around by
rearrangements
throughout
the course of evolution.
One very simple w... | 25,955 |
Genome Assembly as Shortest Superstring
Description:
Introduction to Genome Sequencing
Figure 1
.
Fragment Assembly works by blasting many copies of the same genome into smaller, identifiable reads, which are then used to computationally assemble one copy of the genome.
Recall from
“Computing GC Content”
that... | 25,956 |
Perfect Matchings and RNA Secondary Structures
Description:
Introduction to RNA Folding
Figure 1
.
A hairpin loop is formed when consecutive elements from two different regions of an RNA molecule base pair. (Courtesy: Sakurambo, Wikimedia Commons User)
Because
RNA
is single-stranded, you may have wondered if ... | 25,957 |
Partial Permutations
Description:
Partial Gene Orderings
Similar species will share many of the same
genes
, possibly with modifications.
Thus, we can compare two
genomes
by analyzing the orderings of their genes,
then inferring which
rearrangements
have separated the genes.
In
“Enumerating Gene Orders”
, we... | 25,958 |
Introduction to Random Strings
Description:
Modeling Random Genomes
We already know that the
genome
is not just a random strand of nucleotides; recall from
“Finding a Motif in DNA”
that
motifs
recur commonly across individuals and species. If a
DNA
motif occurs
in many different organisms, then chances are ... | 25,959 |
Enumerating Oriented Gene Orderings
Description:
Synteny Blocks Have Orientations
In
“Enumerating Gene Orders”
, we introduced
synteny blocks
for two different species, which are very
similar areas of two species
genomes
that have been flipped and moved around by
rearrangements
. In that problem, we used the ... | 25,960 |
Finding a Spliced Motif
Description:
Motifs Are Rarely Contiguous
In
“Finding a Motif in DNA”
, we searched for occurrences of a
motif
as a
substring
of a larger
database
genetic string
. However, because a DNA strand coding for a
protein
is often interspersed with
introns
(see
“RNA Splicing”
), we need a... | 25,961 |
Transitions and Transversions
Description:
Certain Point Mutations are More Common
Figure 1
.
Illustration of transitions and transversions.
Point mutations
occurring in
DNA
can be divided into two types:
transitions
and
transversions
.
A transition substitutes one
purine
for another (
$\textrm{A} \left... | 25,962 |
Completing a Tree
Description:
The Tree of Life
Figure 1
.
A phylogeny illustrating proposed evolutionary relationships among the three domains of life: Bacteria, Archaea, and Eukaryota.
"As buds give rise by growth to fresh buds, and these, if vigorous, branch out and
overtop on all sides many a feebler branch... | 25,963 |
Catalan Numbers and RNA Secondary Structures
Description:
The Human Knot
Figure 1
.
Knot fun. Courtesy El Photo Studio.
Figure 2
.
This pseudoknot was formed when bonding occurred at the endpoints of overlapping intervals [1,3] and [2, 4]. (Courtesy: Sakurambo, Wikimedia Commons User)
You may have had the ... | 25,964 |
Error Correction in Reads
Description:
Genome Sequencing Isn't Perfect
In
“Genome Assembly as Shortest Superstring”
, we introduce the problem of assembling a
genome
from a collection
of
reads
. Even though
genome sequencing
is a multi-billion dollar enterprise,
sequencing machines that identify reads still p... | 25,965 |
Counting Phylogenetic Ancestors
Description:
Culling the Forest
Figure 1
.
Trees come in lots of different shapes.
In
“Completing a Tree”
, we introduced the
tree
for the purposes of constructing
phylogenies
.
Yet the definition of tree as a connected graph with no cycles produces a huge class
of different ... | 25,966 |
k-Mer Composition
Description:
Generalizing GC-Content
Figure 1
.
The 2-mer composition of TTGATTACCTTATTTGATCATTACACATTGTACGCTTGTGTCAAAATATCACATGTGCCT
A length
$k$
substring
of a
genetic string
is commonly called a
k-mer
.
A genetic string of length
$n$
can be seen as composed of
$n-k+1$
overlapping... | 25,967 |
Speeding Up Motif Finding
Description:
Shortening the Motif Search
In
“Finding a Motif in DNA”
, we discussed the problem of searching a
genome
for a known
motif
.
Because of the large scale of
eukaryotic
genomes, we need to accomplish this computational task as
efficiently as possible.
The standard method f... | 25,968 |
Finding a Shared Spliced Motif
Description:
Locating Motifs Despite Introns
In
“Finding a Shared Motif”
, we discussed searching through a database containing multiple
genetic strings
to find a
longest common substring
of these strings, which served
as a
motif
shared by the two strings. However, as we saw in... | 25,969 |
Ordering Strings of Varying Length Lexicographically
Description:
Organizing Strings of Different Lengths
In
“Enumerating k-mers Lexicographically”
, we introduced the
lexicographic order
for
strings
of the same
length constructed from some ordered underlying
alphabet
.
However, our experience with dictionarie... | 25,970 |
Maximum Matchings and RNA Secondary Structures
Description:
Breaking the Bonds
In
“Perfect Matchings and RNA Secondary Structures”
, we considered a problem that required us to assume that every possible
nucleotide
is involved in
base pairing
to induce an
RNA
secondary structure
.
Yet the only way this could... | 25,971 |
Creating a Distance Matrix
Description:
Introduction to Distance-Based Phylogeny
A number of different approaches are used to build
phylogenies
, each one featuring its own computational
strengths and weaknesses. One of these measures is
distance-based phylogeny
,
which constructs a
tree
from evolutionary dista... | 25,972 |
Reversal Distance
Description:
Rearrangements Power Large-Scale Genomic Changes
Perhaps the most common type of
genome rearrangement
is an
inversion
,
which flips an entire interval of DNA found on the same
chromosome
.
As in the case of calculating
Hamming distance
(see
“Counting Point Mutations”
),
we would... | 25,973 |
Matching Random Motifs
Description:
More Random Strings
In
“Introduction to Random Strings”
, we discussed searching for
motifs
in large
genomes
, in which
random occurrences of the motif are possible. Our aim is to quantify just how frequently
random motifs occur.
One class of motifs of interest are
promot... | 25,974 |
Counting Subsets
Description:
Characters and SNPs
A
character
is any feature (genetic, physical, etc.) that divides a collection of organisms
into two separate groups. One commonly used genetic character is the possession of a
single-nucleotide polymorphism
, or SNP.
In a process called
genotyping
, the SNP
m... | 25,975 |
Introduction to Alternative Splicing
Description:
The Baby and the Bathwater
Figure 1
.
Alternative splicing induces different protein isoforms.
In
“RNA Splicing”
, we described the process by which the
exons
are spliced out from
a molecule of
pre-mRNA
and reassembled to yield a final
mRNA
for the purpos... | 25,976 |
Edit Distance
Description:
Point Mutations Include Insertions and Deletions
In
“Counting Point Mutations”
, we saw that
Hamming distance
gave us a preliminary
notion of the evolutionary distance between two
DNA strings
by counting the minimum number of single
nucleotide
substitutions that
could have occurred ... | 25,977 |
Expected Number of Restriction Sites
Description:
A Shot in the Dark
In
“Locating Restriction Sites”
, we first familiarized ourselves with
restriction enzymes
.
Recall that these enzymes are used by bacteria to cut through both strands of viral
DNA
,
thus disarming the virus: the viral DNA locations where these ... | 25,978 |
Motzkin Numbers and RNA Secondary Structures
Description:
The Dirty Truth About Mathematics Parties
In
“Catalan Numbers and RNA Secondary Structures”
, we talked about counting the number of ways for an even number of people to
shake hands at a party without crossing hands. However, in the real world, parties only... | 25,979 |
Distances in Trees
Description:
Paths in Trees
For any two
nodes
of a
tree
, a unique
path
connects the nodes;
more specifically, there is a unique path connecting any pair of
leaves
.
Why must this be the case? If more than one path connected two nodes, then they would
necessarily form a cycle, which would v... | 25,980 |
Interleaving Two Motifs
Description:
Two Motifs, One Gene
Recall that in
“Finding a Shared Spliced Motif”
, we found the longest
motif
that could have been shared by two
genetic strings
,
allowing for the motif to be split onto multiple
exons
in the process. As a result,
we needed to find a
longest common su... | 25,981 |
Introduction to Set Operations
Description:
Forming New Sets
Just as numbers can be added, subtracted, and multiplied, we can manipulate
sets
in certain
basic ways. The natural operations on sets are to combine their
elements
, to find those
elements common to both sets, and to determine which elements belong to... | 25,982 |
Sorting by Reversals
Description:
Reconstructing Evolutionary Histories
When we calculate the
Hamming distance
between two
genetic strings
, we can easily
infer a collection of
point mutations
that occurred on the evolutionary path between
the two strings by simply examining the mismatched symbols. However, wh... | 25,983 |
Inferring Protein from Spectrum
Description:
Introduction to Mass Spectrometry
In
“Calculating Protein Mass”
, we briefly mentioned an analytic chemical method called
mass spectrometry
,
which aims to measure the mass-to-charge ratio of a particle or a molecule.
In a mass spectrometer, a sample is vaporized (turne... | 25,984 |
Introduction to Pattern Matching
Description:
If At First You Don't Succeed...
We introduced the problem of finding a
motif
in a
genetic string
in
“Finding a Motif in DNA”
.
More commonly, we will have a collection of motifs that we may wish to find
in a larger string, for example when searching a
genome
for ... | 25,985 |
Comparing Spectra with the Spectral Convolution
Description:
Comparing Spectra
Suppose you have two
mass spectra
, and you want to check if they both were obtained from the
same
protein
; you will need some notion of spectra similarity.
The simplest possible metric would be to count the number of peaks in the mass... | 25,986 |
Creating a Character Table
Description:
Introduction to Character-Based Phylogeny
Before the modern genetics revolution,
phylogenies
were constructed from physical
characters
resulting from direct structural comparison of
taxa
. A great deal of analysis relied on the
fossil record, as fossils provided the only... | 25,987 |
Constructing a De Bruijn Graph
Description:
Wading Through the Reads
Because we use multiple copies of the
genome
to generate and identify
reads
for the purposes
of
fragment assembly
, the total length of all reads will be much longer than the genome itself.
This begs the definition of
read coverage
as the av... | 25,988 |
Edit Distance Alignment
Description:
Reconstructing Edit Distance
In
“Counting Point Mutations”
, the calculation of
Hamming distance
gave us a clear way to
model the sequence of
point mutations
transforming one
genetic string
into another. By simply writing one string directly over the other, we could count... | 25,989 |
Inferring Peptide from Full Spectrum
Description:
Ions Galore
In
“Inferring Protein from Spectrum”
, we inferred a
protein string
from a list of
b-ions
. In practice, biologists have no way
of distinguishing between b-ions and
y-ions
in the
simplified spectrum
of a
peptide
.
However, we will often possess a... | 25,990 |
Independent Segregation of Chromosomes
Description:
Mendel's Work Examined
Mendel's laws of
heredity
were initially ignored, as
only 11 papers have been found that cite his paper between its publication
in 1865 and 1900. One reason for Mendel's lack of popularity is that information
did not move quite so readily ... | 25,991 |
Finding Disjoint Motifs in a Gene
Description:
Disjoint Motifs
In this problem, we will consider an algorithmic (but not particularly practical)
variant of
motif
finding for multiple motifs.
Say we have two motifs corresponding to the
coding regions
of
genes
,
and we want to know whether these motifs can be fou... | 25,992 |
Finding the Longest Multiple Repeat
Description:
Long Repeats
We saw in
“Introduction to Pattern Matching”
that a
data structure
commonly used to encode the relationships
among a collection of
strings
was the
trie
, which is particularly useful when
the strings represent a collection of
patterns
that we wis... | 25,993 |
Newick Format with Edge Weights
Description:
Weighting the Tree
A vital goal of creating
phylogenies
is to quantify a molecular clock that indicates
the amount of evolutionary time separating two members of the phylogeny. To this end, we will
assign numbers to the
edges
of a
tree
so that the number assigned t... | 25,994 |
Wobble Bonding and RNA Secondary Structures
Description:
Don't Look Down
We have discussed the problem of counting
RNA
secondary structures
in previous problems. In this problem, we will add some assumptions to those
used in
“Motzkin Numbers and RNA Secondary Structures”
to provide ourselves with an ultimatel... | 25,995 |
Counting Disease Carriers
Description:
Genetic Drift and the Hardy-Weinberg Principle
Mendel's laws of
segregation
and
independent assortment
are excellent for the study of individual organisms and their progeny,
but they say nothing about how
alleles
move through a population over time.
Our first question is:... | 25,996 |
Creating a Character Table from Genetic Strings
Description:
Phylogeny from Genetic Characters
In
“Creating a Character Table”
, we introduced the
character table
as a way of representing a number
of
characters
simultaneously. In that problem, we found a character table representing
an
unrooted binary tree
o... | 25,997 |
Counting Optimal Alignments
Description:
Beware of Alignment Inference
In
“Edit Distance Alignment”
, we introduced the concept of an
alignment
of two
genetic strings
having differing lengths with respect to
edit distance
.
This provided us with a way of visualizing a specific collection of
symbol substitution... | 25,998 |
Counting Unrooted Binary Trees
Description:
Counting Trees
A natural question is to be able to count the total number of
distinct
unrooted binary trees
having
$n$
leaves, where each leaf is labeled by some
taxon
. Before we can count
all these trees, however, we need to have a notion of when two such trees a... | 25,999 |
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