# Triangular Prism Tutorial

## Introduction to Prisms and Triangular Prisms

A prism is a polyhedron with an n-sided polygonal base, its translated copy, and n faces joining corresponding sides. Specifically, a Triangular Prism is a polyhedron consisting of a triangular base, its translated copy, and 3 faces joining corresponding sides.

## Key Formulas for Triangular Prisms

- **Area of Base (A)**: ½ * a * b, where a = altitude and b = base.
- **Perimeter of Base (P)**: s1 + s2 + s3, where s1, s2, and s3 are the sides of the base.
- **Surface Area of Prism**: ab + (s1 + s2 + s3)h = ab + Ph, where h = height.
- **Volume of Prism**: ½ * a * b * h = Ah.

## Example Calculation

Given a triangular prism with altitude (a) = 2, base (b) = 3, height (h) = 4, and sides s1 = 1, s2 = 2, s3 = 3, calculate its surface area and volume.

1. **Find the Area of the Base**: A = ½ * a * b = 0.5 * 2 * 3 = 3.
2. **Find the Perimeter of the Base**: P = s1 + s2 + s3 = 1 + 2 + 3 = 6.
3. **Calculate the Surface Area**: Surface Area = ab + Ph = (2 * 3) + (6 * 4) = 6 + 24 = 30.
4. **Calculate the Volume**: Volume = Ah = 3 * 4 = 12.

This tutorial provides a step-by-step guide on calculating the volume, surface area, and perimeter of a triangular prism, aiding in the understanding and solution of math area problems related to triangular prisms.


# Answer Probability Questions with Simulation

Probability is a fundamental concept in data science, and simulation is a commonly used technique in algorithms such as the bootstrap. This exercise set aims to strengthen intuition for probability and simulation algorithms.

The problems in this set have exact analytical solutions, making them ideal for practice. To get the most out of the exercises, read the instructions carefully and think about the solution before writing R code.

**Exercise 1**
In 100 coin tosses, what is the probability of having the same side come up 10 times in a row? Useful functions: `sample(), rbinom(), rle()`.

**Exercise 2**
Six kids are standing in line. What is the probability that they are in alphabetical order by name, assuming no two children have the same exact name?

**Exercise 3**
With three boys and three girls, what is the probability that all the girls come first?

**Exercise 4**
In six coin tosses, what is the probability of having a different side come up with each throw, with no consecutive tails or heads?

**Exercise 5**
What is the probability of a flush (all cards are the same suit) in a random five-card poker hand from a standard deck?

**Exercise 6**
In a random thirteen-card hand from a standard deck, what is the probability that none of the cards is an ace and none is a heart?

**Exercise 7**
At four parties with 13, 23, 33, and 53 people, what is the probability that at least two individuals share a birthday at each party, assuming no leap days and uniform birth distribution?

**Exercise 8**
A coin tossing game has a stake that doubles with each heads, starting at $2. The game ends when a tail appears or after 1000 tosses. What is the probability of profit if participation costs $15?

**Exercise 9**
What is the probability that the pattern heads-heads-tails appears before tails-heads-heads in coin tossing?

**Exercise 10**
In a game show with three doors, one with a car and two with goats, what is the probability of winning the car if you pick a door, the host opens another door with a goat, and you switch doors?


**Problem 1: Atmospheric Carbon Dioxide**

Atmospheric carbon dioxide is thought to be responsible for global warming. The table below contains data about the amount of Atmospheric CO2 (in ppm) over the last fifty years.

| Year | CO2 (ppm) |
| --- | --- |
| 1959 | 315.98 |
| 1960 | 316.91 |
| 1965 | 320.04 |
| 1968 | 323.04 |
| 1971 | 326.32 |
| 1975 | 331.08 |
| 1980 | 338.7 |
| 1982 | 341.4 |
| 1984 | 344.4 |
| 1986 | 347.2 |
| 1988 | 351.5 |
| 1990 | 354.2 |
| 1992 | 356.4 |
| 1994 | 358.9 |
| 1996 | 362.6 |
| 1998 | 366.6 |
| 2000 | 369.4 |
| 2002 | 372.9 |
| 2007 | 383.71 |
| 2008 | 385.54 |
| 2009 | 387.35 |

(a) Construct a scatter plot for this data.
(b) Comment on the shape of the scatter plot.
(c) Find a line of best fit for the data.
(d) Using the equation, estimate the atmospheric CO2 in 2005, 1920, and 2020. Comment on the accuracy of your answers.
(e) Comment on the accuracy of modeling this as a straight line given your graph.

**Problem 2: Forensic Science – Bone Lengths**

Forensic scientists can estimate the height of deceased people from the length of a few key bones. The equations for estimating height from bone length are:

* For males: H = 3.08h + 70.45 (humerus), H = 2.42t + 81.93 (tibia), H = 3.7u + 70.45 (ulna)
* For females: H = 3.36h + 57.97 (humerus), H = 2.9t + 61.53 (tibia), H = 4.27u + 57.76 (ulna)

(a) A tibia of length 32 cm is found in a burial site. Estimate the person's height.
(b) A collection of bones is found with the following lengths: humerus = 34 cm, tibia = 38.5 cm, and ulna = 28.3 cm. Determine the person's sex and height.

**Problem 3: Tyre Life**

Data about tyre wear was collected for two brands of tyres. The number of kilometres for each brand is:

* Brand A: 35,000, 38,000, 38,000, 42,000, 45,000, 45,000, 49,000, 50,000, 50,000, 52,000, 53,000, 58,000, 62,000, 62,000, 63,000, 66,000, 67,000, 70,000, 71,000, 74,000, 76,000, 81,000, 81,000, 86,000, 94,000
* Brand B: 46,000, 49,000, 55,000, 56,000, 58,000, 64,000, 64,000, 64,000, 67,000, 68,000, 68,000, 69,000, 69,000, 72,000, 72,000, 74,000, 76,000, 78,000, 79,000, 79,000, 82,000, 86,000, 87,000, 94,000, 97,000

(a) Draw a side-by-side box and whisker plot to compare the two brands.

**Problem 4: Male and Female Athletes**

The table below contains times for males and females in 100m running races at the Olympic Games.

| Year | Years since 1928 | Men's 100m time (sec) | Women's 100m time (sec) |
| --- | --- | --- | --- |
| 1928 | 0 | 10.8 | 12.2 |
| 1932 | 4 | 10.3 | 11.9 |
| 1936 | 8 | 10.3 | 11.5 |
| 1948 | 20 | 10.3 | 11.9 |
| 1952 | 24 | 10.4 | 11.5 |
| 1960 | 32 | 10.2 | 11.0 |
| 1968 | 40 | 9.95 | 11.08 |
| 1976 | 48 | 10.06 | 11.08 |
| 1980 | 52 | 10.25 | 11.06 |
| 1992 | 64 | 9.96 | 10.82 |
| 1996 | 68 | 9.84 | 10.94 |
| 2000 | 72 | 9.87 | 10.75 |
| 2004 | 76 | 9.85 | 10.93 |
| 2008 | 80 | 9.69 | 10.78 |

(a) Construct a graph for this data.
(b) Determine the equation for the line of best fit by both calculation and using Excel. Compare the two answers.
(c) Will female athletes ever catch up to male athletes? When might this happen?

**Problem 5: Household Water Use**

(a) During a recent rainy day, 72 mm of rain was recorded. Calculate the volume of water collected from the roof and stored in the water tank.
(b) The volume V of water in the tank, initially full, remaining after d days is given by the equation: V = 10,000 - 235d. Calculate the amount of water left after 30 days of no rain.
(c) Determine the total capacity of the water tank.
(d) Determine the daily water usage of the household.
(e) When will the volume of the tank be reduced to 2500 litres?

**Problem 6: Break-even Analysis**

A tourist coach company offers a daily tour from Byron Bay to Lismore and return. The costs and revenue are:

* Weekly advertisement: $120
* Fuel cost and other running costs: $45 per trip
* Driver's wage: $120 per trip
* Lunch cost: $14 per person
* Revenue per passenger: $45

(a) Plot the graphs of revenue and cost and find the number of passengers required per week to break even.
(b) Determine the number of passengers required to make a profit of $1000 per week.
(c) What is the maximum profit possible if the bus can hold a maximum of 20 passengers per trip?


## Homework Statement
An elevator and its load have a mass of 1200 kg.

a) What energy must be supplied to lift the elevator and its load to a height of 100m?
Answer: E = _________ Joules

b) If the elevator rises at a speed of 5 m/s, how long does it take to cover 100m?
Answer: t = __________ sec

c) What is the power supplied?
Answer: P = ________ Watts

## Homework Equations
Energy: E = MgL, where g = 9.8 m/s^2 is the acceleration of gravity and L is the distance through which a mass is moved against the force of gravity
Power: P = E/t

## Solution Attempt
For part a, using E = MgL, we get E = 1200 kg * 9.8 m/s^2 * 100 m = 1176000 Joules.

For part b, at a rate of 5 m/s, it would take t = 100 m / 5 m/s = 20 seconds.

For part c, using P = E/t, we get P = 1176000 Joules / 20 seconds = 58800 Watts.

Given answers for verification:
a) E = 1176000 Joules
b) t = 20 seconds
c) P = 58800 Watts


To express a fraction as a percentage, the numerator of the fraction is used when the denominator is 100. For example, 11% is equivalent to 11/100 as a fraction.

Converting a fraction to a percentage involves dividing the fraction by 100. Conversely, to convert a percentage to a fraction, the percentage is expressed as the numerator over 100.

For instance, 0.5 is equivalent to 1/2 as a fraction and 50% as a percentage. Similarly, 1.23 can be expressed as 123/100 as a fraction and 123% as a percentage. The fraction 3/5 is equivalent to 60% as a percentage.

To simplify a percentage into a fraction, it is expressed as the numerator over 100. For example, 0.000009% is equivalent to 9/100000000 as a fraction.

In summary, fractions and percentages are interchangeable, with the percentage being the numerator of an equivalent fraction with a denominator of 100. Examples include:
- 0.5 = 1/2 = 50%
- 1.23 = 123/100 = 123%
- 3/5 = 60%
- 11% = 11/100
- 0.000009% = 9/100000000

Q: What is the percentage for the fraction half?
A: The percentage for the fraction half is 50%, since half is equivalent to 1/2 as a fraction and 0.5 as a decimal.

Given that 0.5 is half, it can be expressed as a percentage (50%) and as a fraction (1/2). To express a number as a percentage, it is converted to a fraction with a denominator of 100. For example, 35 as a fraction is 35/1, and 3500% as a percentage.


**Stock Lists in Excel**

Handling and keeping stock data is crucial as it holds the total profit or loss. The profit or loss of any stock can be called as square off or spread. Each stock is bought on two basis: Quantity and price of each stock. There are two ways to get the total spread or square off.

**Calculating Stock Prices**

To calculate the total stock price, multiply the quantity and price/stock and take the sum for different stocks. This can be done using the PRODUCT formula. For example, to calculate the total stock price spent on ACC, use the formula: `=PRODUCT(B4,C4)`. This formula can be copied and pasted to further cells by using Ctrl + C and Ctrl + V or by drag and drop.

**Using the SUMPRODUCT Formula**

The SUMPRODUCT formula gives the total bought price and current price for the bought stocks. To use this formula, select the range of cells containing the quantity and price/stock data. For example, to calculate the total stock bought price, use the formula: `=SUMPRODUCT(B4:B8,C4:C8)`. This formula multiplies the corresponding values in the two arrays and returns the sum of the products.

**Example**

Suppose we have 5 stocks with their quantity, bought price/stock, and current price/stock. To calculate the total stock price spent on each stock, use the PRODUCT formula. Then, to calculate the total current price of the bought stocks, use the same PRODUCT formula with the current price/stock data. Finally, to get the total profit, subtract the total bought price from the total current price.

**Notes**

* The SUMPRODUCT formula only works for numbers.
* The length of arrays in the SUMPRODUCT function must be the same, or else the function throws an error.

**Alternative Solution**

If you need to know the statistics for all different stocks, use the PRODUCT formula for each stock. However, if you need to know the statistics of the whole data, use the SUMPRODUCT formula.

**Related Articles**

* How to use the SUMPRODUCT function in Excel
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* 2 Ways to Sum by Month in Excel
* How to Sum Multiple Columns with Condition
* How to use wildcards in Excel

**Popular Articles**

* 50 Excel Shortcuts to Increase Your Productivity
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**Tips and Tricks**

* Use the shortcut key Alt+= to autosum in MS Excel.
* Use the SUMPRODUCT formula to save file space and make calculations easier to understand.


Write an Equation Given the Slope and a Point

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The slope can be determined using the formula m = (y2 - y1) / (x2 - x1), given the coordinates of two points on the line (x1, y1) and (x2, y2). The y-intercept, b, is the point at which the line crosses the y-axis.

To find the equation of a line given the slope and a single point, two methods can be used. For example, given a slope of m = 3/4 and a point on the line (4, 5), the equation can be found as follows:

Method 1: Using the slope-intercept equation, plug in the values to solve for b.
5 = (3/4)(4) + b
5 = 3 + b
b = 2
The slope-intercept equation is y = (3/4)x + 2.

Method 2: Using the point-slope formula, (y - y1) = m(x - x1), plug in the information and simplify.
(y - 5) = (3/4)(x - 4)
(y - 5) = (3/4)x - 3
y = (3/4)x - 3 + 5
y = (3/4)x + 2

The slope-intercept formula is y = mx + b, where m is the slope and b is the y-intercept. The slope is the change in y over the change in x, also known as the rise over run or Δy / Δx. If two points on the line are known, the slope can be found using the formula (y2 - y1) / (x2 - x1).

The y-intercept is the point where the line touches or intersects the y-axis. If the x-coordinate is zero, the y-intercept can be found. The x-intercept is the value of x where y = 0, also known as the root of the equation.

Given the slope and x-intercept, the equation of a line can be found. For example, given a slope of 2 and an x-intercept of 3, the equation can be found as follows:
0 = 2(3) + b
b = -6
The equation is y = 2x - 6.

Questions:

Forms of Linear Equations

1. Write an Equation Given the Slope and a Point
2. Write an Equation Given Two Points
3. Write a Function in Slope-Intercept Form
4. Linear Equations in Point-Slope Form
5. Forms of Linear Equations
6. Applications Using Linear Models
7. Equations of Parallel Lines
8. Equations of Perpendicular Lines
9. Families of Lines
10. Fitting Lines to Data
11. Linear Interpolation and Extrapolation
12. Problem Solving with Linear Models
13. Dimensional Analysis


In Maths, a polynomial is a concept studied under the Algebra branch. To understand polynomials, we first need to define basic terms. Variables are symbols that can be assigned different numerical values, such as \(x, y, z, p, q, r, s\), etc. A symbol with a fixed value is called a constant, for example, \(8, 5, 9, \pi\), etc.

A combination of constants and variables connected by basic operations \(+, -, \times, \div\) is called an algebraic expression. For instance, \(7+8x\) and \(7+8x-6x^2y-{\frac{4}{9}xy}\) are algebraic expressions. The parts of an algebraic expression separated by \(+\) or \(-\) operations are called terms. In the expression \(7+8x-6x^2y-{\frac{4}{9}xy}\), the terms are \(7\), \(8x\), \(-6x^2y\), and \(-\frac{4}{9}xy\).

An algebraic expression with variables having non-negative integral powers is called a polynomial. For example, \(x^3+2x^2+5x+7\) and \(x^2y+xy^2+7y\) are polynomials because the powers of the variables are non-negative integers.

The degree of a polynomial with one variable is the highest power of the variable. In the polynomial \(3x^4+5x^3+7x^2+8\), the degree is 4 because the highest power of \(x\) is 4.

Polynomials are classified based on the degree and the number of terms. The types of polynomials include:
- Monomial: a polynomial with one non-zero term, such as \(8x^5\) or \(-7y^3\).
- Binomial: a polynomial with two non-zero terms, such as \(7x^6-3x^4\).
- Trinomial: a polynomial with three non-zero terms, such as \(5x^4+3x^2-8\).
- Quadrinomial: a polynomial with four terms, such as \(7x^5-3x^2+9x+5\).
- Quintrinomial: a polynomial with five terms, such as \(y^6+8y^5+9y^4+9y^2+7\).

Polynomials can also be classified based on their degree:
- Linear polynomial (degree 1): \(ax+b\), such as \(x^1+8\).
- Quadratic polynomial (degree 2): \(ax^2+bx+c\), such as \(5x^2+2x-7\).
- Cubic polynomial (degree 3): \(ax^3+bx^2+cx+d\), such as \(2x^3+3x^2-5x+7\).
- Biquadratic polynomial (degree 4): \(ax^4+bx^3+cx^2+dx+e\), such as \(2x^4-3x^2+5x^3+7x-4\).
- Constant polynomial (degree 0): a polynomial that can be written as \(ax^0\), such as \(4\) or \(-7\).

A polynomial with all coefficients equal to zero is called a zero polynomial, and its degree is not defined. A polynomial is in standard form if all variables are in ascending or descending order. The standard form of a polynomial is \(a_0 + a_1x + a_2x^2 + \cdots + a_nx^n\), where \(a_0, a_1, a_2, \ldots, a_n\) are real numbers and \(n\) is a whole number.

The coefficient of the highest power term is called the leading coefficient, and the term with the highest exponent is called the leading term. For example, in the polynomial \(8x^4 + 6x^3 + 8x^2 - 4x + 2\), the leading coefficient is 8, and in the polynomial \(8x^4 + 3x^3 + 2x^2 - 7x + 6\), the leading term is \(8x^4\).


The 2014 World Surfing League data from surf-stats.com contains average scores for 55 surfers under various conditions, including wind, wave size, surf break, and weather. Scores range from 0 to 20, with each surfer aiming to achieve their two highest scoring waves, both out of a possible 10 points, for a total of 20 points per heat. A panel of five judges scores each wave, discounting the highest and lowest scores, and the surfer receives the average of the remaining three scores.

To visualize surfer performance, correlations between scores can be examined in a scatterplot. However, this approach can be messy and lacking in useful information. Principal Component Analysis (PCA) can reduce the number of dimensions while capturing important information. In R, the `princomp` function creates principal components from a data matrix.

Applying PCA to the surfing data results in 22 components, with the first two components capturing 69% of the variance. The importance of these components is as follows:

* Component 1: standard deviation of 3.5987, proportion of variance of 0.5887, cumulative proportion of 0.5887
* Component 2: standard deviation of 1.4812, proportion of variance of 0.0997, cumulative proportion of 0.6884

Plotting each surfer on a scatterplot using Component 2 (x-axis) and Component 1 (y-axis) reveals that top surfers, such as Gabriel Medina and Kelly Slater, are located at the bottom center of the plot, while lower-ranked surfers are spread across the top.

To improve, a surfer like Jordy Smith, who finished 6th in 2014 with a total score of 721.61, should focus on the loadings of each component. The factor loadings analysis reveals that Jordy should improve his score on small waves (1-4 ft), overcast weather conditions, test conditions, right wave direction, and beach breaks to move downwards in the plot. Additionally, improving his score in reef breaks would move him towards the left of the plot.

The key variables for improvement are:
* Small waves (1-4 ft): -0.2374
* Overcast weather: -0.2367
* Test conditions: -0.2354
* Right wave direction: -0.2301
* Beach breaks: -0.2301
* Reef breaks: -0.2019

By focusing on these areas, Jordy Smith can close the gap with top surfers like Medina and improve his overall performance.


**Differentiated Learning Objectives**

The learning objectives for students are as follows:
1. All students should be able to measure the area of a rectangle by counting the squares inside.
2. Most students are expected to calculate the area of a rectangle as a product of its length and width.
3. Some students should be able to express the area as a mathematical expression.


Part 1: Starting from Zero

Part 2: Meet Eustace

Part 3: Bernoulli Trials

Part 4: Why Randomness is Not All Equally Random

Probability theory has a term for plotting out a life or a random process: a *strong solution*. A strong solution is a specified trajectory for a random process, such as the realized sequence of heads and tails in a coin flipping game. However, Eustace can't know this path in advance and can only construct a distribution of possible outcomes, known as a *weak solution*. A weak solution is defined by the moments of a probability distribution and is specified by the expectations or moments of the underlying random process.

If Eustace knows a random process is stationary, he has confidence that the moments of the process will converge to the same values every time. For instance, the coin flipping game is stationary, and its long-term average winnings will always converge to zero. Eustace is limited to a weak solution when looking into an uncertain future, and the actual path remains a mystery.

A weak solution is fine for mathematics, but from Eustace's point of view, it's no solution at all. Sooner or later, he must risk a strong solution and chart a course. The accumulated information on which Eustace can base his course is called a *filtration*. A filtration represents all information available to Eustace when he chooses a course of action and is technically an algebra defined on an increasing set of information.

In the coin flipping game, Eustace's filtration includes the record of previous flips and the constraints of the game itself. The filtration helps him predict his overall wins and losses, but not the next flip. If Eustace wins the first flip, he knows that after the next flip, he can't be negative. This is more information than he had when he started.

The expected value of Eustace's wealth at any point is simply what he has won up to that point. The past reveals what Eustace has won, and the future of this stationary distribution is defined by unchanging moments. In a fair game, Eustace can expect to make no money, no matter how lucky he feels.

Two simple equations summarize this:

**E(wealth@t | F@t-1) = wealth@t-1** (first moment)

**variance(wealth@t|F@t-1) = 1** (second moment)

The expected wealth at any time t is the wealth Eustace has accumulated up until time t-1. The filtration, F, accumulates the set of distinguishable events in the realized history of a random process. In the coin game, the outcome of each flip adds information to Eustace's filtration.

There is always a termination point, though it need not always be zero; maybe Eustace needs to save a few bucks for the bus ride home. Let's call this point *wealth_c* (critical). Introducing this term into the original equation for expected wealth, we now have:

**max E(wealth@t – wealth_c | F@t-1)**

Eustace's thermodynamic environment works the same way. In the casino, he can't blindly apply any particular strong solution. Each card dealt in each hand will influence his subsequent actions in accordance with the content of his filtration. The best strategy is always the one with **max E(wealth@t|F@t-1)** at each turn.

As Eustace graduates to higher levels of the casino, the games become more complex. He needs some way of accommodating histories, and inflexibility is a certain path to ruin. Card-counters differ from suckers at blackjack only by employing a more comprehensive model that adapts to the available filtration. They act on more information and can apply the optimal strong solution every step of the way.

In the alpha casino, Eustace encounters myriad random processes. His ability to mediate the effects of these interactions is a direct consequence of the configuration of his alpha model. The best he can hope to do is accommodate as much of the filtration into this model as he can to generate the best possible response. Suboptimal responses will result in smaller gains or greater losses of alpha.

**Disclaimer**: Although the language of agency is used, nothing in this discussion is intended to impute agency, or consciousness, or even life, to Eustace. One could speak of any inanimate object with a feedback mechanism in exactly the same way.


The **Friedman Test** is a non-parametric alternative to the Repeated Measures ANOVA, used to determine if there is a statistically significant difference between the means of three or more groups with the same subjects in each group.

**When to Use the Friedman Test**

The Friedman Test is commonly used in two situations:
1. **Measuring mean scores over time**: For example, measuring the resting heart rate of subjects at multiple time points (e.g., before, during, and after a training program).
2. **Measuring mean scores under different conditions**: For example, having subjects rate their enjoyment of different movies.

**Friedman Test Example**

Suppose we want to determine if the mean reaction time of subjects differs on three different drugs. We recruit 10 patients, measure their reaction times on each drug, and use the Friedman Test to compare the mean reaction times.

**Step 1: State the Hypotheses**
* **Null Hypothesis (H0)**: The mean reaction times are equal across the three drugs (µ1 = µ2 = µ3).
* **Alternative Hypothesis (Ha)**: At least one population mean is different from the rest.

**Step 2: Perform the Friedman Test**
Using a Friedman Test Calculator with the input data, we obtain the test statistic and p-value.

**Step 3: Interpret the Results**
The test statistic is Q = 12.35, and the corresponding p-value is p = 0.00208. Since p < 0.05, we reject the null hypothesis, concluding that the type of drug used leads to statistically significant differences in response time.

**Step 4: Report the Results**
A Friedman Test was conducted on 10 patients to examine the effect of three different drugs on response time. Results showed that the type of drug used led to statistically significant differences in response time (Q = 12.35, p = 0.00208).

**Additional Resources**
Tutorials are available on performing the Friedman Test using various statistical software, including Excel, R, Python, and Stata, as well as an online Friedman Test Calculator.


# Marathon

## Problem Description

Farmer John enrolls his cows in physical fitness activities due to their poor health. His prize cow, Bessie, is enrolled in a running class where she will run a marathon through the downtown area. The marathon course consists of N checkpoints (3 <= N <= 500) that must be visited in sequence. Bessie plans to skip up to K checkpoints (K < N) to shorten her journey, but she cannot skip the first or last checkpoint.

The distance between two checkpoints at locations (x1, y1) and (x2, y2) is given by |x1-x2| + |y1-y2|. The goal is to find the minimum distance Bessie must run if she can skip up to K checkpoints.

## Input Format

The input consists of the number of checkpoints N, the number of checkpoints Bessie can skip K, and the coordinates of each checkpoint.

## Output Format

The output is the minimum distance Bessie must run.

## Example Input

5 2 0 0 8 3 1 1 10 -5 2 2

## Example Output

4

## Note

The course is set in a downtown area with a grid of streets, and the distance between checkpoints is calculated using the Manhattan distance formula (|x1-x2| + |y1-y2|).


The base angles in an isosceles trapezoid are equal, formed between non-parallel sides and parallel sides. The diagonals of an isosceles trapezoid are equal in length. The sum of opposite angles in an isosceles trapezoid is 180 degrees.

It is not true that all isosceles trapezoids are similar. For example, two isosceles triangles can have congruent angles within each triangle, but the angles of one are not congruent to the angles of the other, making them non-similar.

The properties of an isosceles trapezoid include having a convex polygon and cyclic shape. Both parallel sides are called bases. In an isosceles trapezoid, the two base angles are congruent, similar to an isosceles triangle. The diagonals of an isosceles trapezoid are congruent, but they do not bisect each other, as stated in the Isosceles Trapezoid Diagonals Theorem.

To determine if diagonals bisect angles in an isosceles trapezoid, the answer is no, they do not. The diagonals are congruent, but they do not intersect at their midpoints, and they do not bisect the angles of the trapezoid.


**Introduction to Public-Key Cryptosystems**

Public-key cryptosystems are widely used for secure communication, with RSA being one of the most popular algorithms. RSA relies on a clever system of public and private keys to ensure secure data transmission. To understand RSA, it's essential to grasp the basics of congruence relations and discrete mathematics.

**A Little History**

The Caesar cipher, developed around 50 BC, is a simple encryption technique where each letter is shifted by three positions in the alphabet. This can be expressed mathematically as a function that assigns to a non-negative integer, where. The encryption process involves replacing each letter with its corresponding integer, applying the function, and then translating the result back to letters.

**Public-Key Cryptography**

Public-key cryptosystems use a pair of keys: a public key for encryption and a private key for decryption. The public key is freely available, while the private key is kept secret. This approach ensures that only the intended recipient can decrypt the message. In contrast, private-key cryptosystems use the same key for both encryption and decryption, which can be a significant weakness.

**Introducing the RSA Cryptosystem**

RSA, developed in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman, is a widely used public-key cryptosystem. It relies on modular exponentiation modulo the product of two large primes, and. The encryption key consists of two integers, and, where is the exponent and is the modulus. The decryption key also consists of two integers: the same modulus and the integer, an inverse of modulo.

**Encryption Phase**

The encryption process involves translating the message into a sequence of integers, grouping them into blocks, and applying the encryption function. The function uses modular exponentiation to transform the integer into an integer. To compute the exponent, Euler's totient function is used, which is multiplicative. The encryption key is then used to encrypt the message.

**Example**

Let's use the message "THE CAKE IS A LIE" with, , and. We translate each letter into its corresponding two-digit integer representation, group them into blocks of four, and apply the encryption function. Using the Right-to-Left Binary method for modular exponentiation, we get the encrypted message.

**Decryption Phase**

To recover the original message, the decryption key is used. The decryption key consists of the same modulus and the integer, an inverse of modulo. The decryption process involves applying the decryption function to each block of integers. Using Fermat's Little Theorem and the Chinese Remainder Theorem, we can show that the decryption key is correct.

**Security Implications**

The security of the RSA cryptosystem is based on the RSA Problem, which involves finding the -th root modulo a composite to recover a value such that. This approach requires factoring the modulus to recover the prime factors and. However, factoring large integers is a computationally infeasible task, making RSA a secure algorithm.

**Conclusion**

RSA is a widely used public-key cryptosystem that relies on modular exponentiation and the difficulty of factoring large integers. Its security is based on the RSA Problem, which makes it a secure algorithm for data transmission. Understanding RSA requires a grasp of discrete mathematics and congruence relations. There are other public-key algorithms, such as the Diffie-Hellman algorithm and the ElGamal signature scheme, which are also worth exploring.


# 10.3. Sand Piles

The sand pile model, proposed by Bak, Tang, and Wiesenfeld in 1987, is an abstraction that models physical systems with a large number of interacting elements. It is a 2-D cellular automaton where each cell represents the slope of a sand pile part. During each time step, cells exceeding a critical value, *K* (usually 3), "topple" and transfer sand to four neighboring cells, decreasing their slope by 4 and increasing each neighbor's by 1. Perimeter cells are kept at slope 0, allowing excess to spill over.

The model is initialized with all cells at a level greater than *K* and run until stabilization. Then, small perturbations are introduced by randomly selecting a cell, incrementing its value by 1, and running the model until it stabilizes again. For each perturbation, `T` (time steps to stabilize) and `S` (total cells that topple) are measured.

Most perturbations cause no toppling, resulting in `T=1` and `S=0`. However, a single grain can occasionally trigger an **avalanche** affecting a substantial fraction of the grid. The distributions of `T` and `S` are heavy-tailed, supporting the claim that the system is in a critical state.

Bak, Tang, and Wiesenfeld conclude that the sand pile model exhibits **self-organized criticality**, meaning it evolves toward a **critical state** without the need for **external control or fine tuning** of parameters. The model remains in a critical state as more grains are added.

Q-1: Bak, Tang and Wiesenfeld conclude that the sand pile model exhibits **self-organized criticality**, which means that it evolves toward a **critical state** without the need for **external control or fine tuning**.


The Simpsons is known for its zany characters, political jokes, and outrageous situations, but some viewers also appreciate its references to mathematics. Several of the show's writers have a background in math or computer science, and they often sneak in mathematical bits.

In the final episode of the 2005-2006 season, a character is asked to guess the attendance of a ball game, with options 8128, 8191, or 8208. These numbers have mathematical significance: 8128 is a perfect number, 8191 is a Mersenne prime, and 8208 is a special four-digit number that can be written as the sum of the fourth powers of its digits.

A perfect number is equal to the sum of its divisors. For example, the smallest perfect number is 6, with divisors 1, 2, and 3, since 1 + 2 + 3 = 6. Other perfect numbers include 28, 496, and 8128.

A Mersenne prime, like 8191, is a prime number that can be written as 2^n - 1, where n is an integer. In this case, 8191 = 2^13 - 1.

The number 8208 has the unique property that (8^4) + (2^4) + (0^4) + (8^4) = 8208. Only two other four-digit numbers, 1634 and 9474, share this property.

Mathematicians Sarah J. Greenwald and Andrew Nestler have tracked math references in The Simpsons on their website, simpsonsmath.com. In 2006, the show even aired an episode devoted to math, called "Girls Just Want to Have Sums." This episode, and others like it, provide another reason to watch The Simpsons, even for educational purposes. For more information on math in The Simpsons, see "Springfield Theory" in Muse, November/December 2006, p. 44.


**Article 10: Finding Solid Feet in a Round Stick of Timber**
To find the solid feet in a round stick of timber, multiply twice the square of its semi-diameter in inches by the length in feet, then divide by 144.

**Example**
Diameter: 22 inches, Length: 20 feet
Calculation: (11x11) x 2 x 20 / 144 = 33.6 solid feet

**Article 11: Finding Feet of Square Edged Boards**
To find the feet of square edged boards, first find the solid content of the log when made square, then use the proportion: As the thickness of the board is to the solid feet, so is 12 inches to the number of feet of boards.

**Example**
Log diameter: 24 inches, Length: 20 feet, Board thickness: 1.7 inches
Calculation: (12x12) x 2 x 20 / 144 = 40 solid feet, then 12 / 1.7 = 384 / 40

**Article 12: Finding Bushels in a Square Box**
To find the bushels in a square box, multiply the length, breadth, and depth, then divide by 2150.425.

**Example**
Length: 50 inches, Breadth: 40 inches, Depth: 60 inches
Calculation: (50 x 40 x 60) / 2150.425 = 55.84 bushels

**Article 13: Finding Bricks Needed for a Wall**
To find the bricks needed, calculate the solid content of the wall, then multiply by the number of bricks per solid foot.

**Example**
Wall dimensions: 44 feet long, 40 feet wide, 20 feet high, 1 foot thick
Brick dimensions: 8 inches long, 4 inches wide, 2 inches thick
Calculation: Find the solid content of the wall, then multiply by the number of bricks per solid foot.

**Article 14: Finding Tonnage of a Ship**
To find the tonnage, multiply the length of the keel by the breadth of the beam by the depth of the hold, then divide by 95.

**Example**
Length: 72 feet, Breadth: 24 feet, Depth: 12 feet
Calculation: (72 x 24 x 12) / 95 = 218.2 tons

**Article 15: Finding Strength of a Cable**
The strength of cables is proportional to the cube of their peripheries.

**Examples**
1. Cable diameter: 6 inches, Anchor weight: 21 cwt, New cable diameter: 12 inches
Calculation: (6x6x6) / 21 = (12x12x12) / x, x = 18 cwt
2. Cable diameter: 12 inches, Anchor weight: 18 cwt, New anchor weight: 2.25 cwt
Calculation: (18 / (12x12x12)) = (2.25 / (xcxxc)), x = 6 inches

**Article 16: Finding Burthen of a Ship**
The burthens of similar ships are proportional to the cubes of their like dimensions.

**Example**
Ship 1: 75 feet long, 300 tons, Ship 2: 100 feet long
Calculation: (75x75x75) / 300 = (100x100x100) / x, x = 711.2 tons

**Duodecimals**
A method for calculating the contents of work using a base-12 system.

**Rule**
1. Write the multiplicand and corresponding denominations of the multiplier.
2. Multiply each term into the multiplicand, carrying units for every 12.
3. Repeat for each denomination in the multiplier, shifting the result to the right.

**Example**
Multiply 9 feet 8 inches 6 seconds by 7 feet 9 inches 3 seconds.
Calculation:
- Multiply by feet: 9 x 7 = 63
- Multiply by inches: 8 x 9 = 72, 6 x 7 = 42
- Multiply by seconds: 6 x 9 = 54, 8 x 3 = 24, 6 x 7 = 42
Result: 63 feet, 72 inches, 54 seconds + 42 inches, 24 seconds + 42 inches, 42 seconds

**Finding Square Feet in a Board**
Length: 16 feet 9 inches, Width: 2 feet 3 inches
Calculation: (16.75 x 2.25) = 37.69 square feet


To find the number of square centimeters in one square meter, we need to understand the relationship between meters and centimeters. Since 1 meter equals 100 centimeters, 1 square meter (m²) is calculated as 100 cm * 100 cm = 10,000 cm².

Therefore, there are 10,000 cm² in one m². This relationship can be used to convert between square meters and square centimeters. To convert m² to cm², you multiply by 10,000, and to convert cm² to m², you divide by 10,000.

For example, to find out how many m² are in 1300 cm², you divide 1300 by 10,000, which equals 0.13 m². Similarly, 0.01 m² equals 0.01 * 10,000 = 100 cm², and 170 cm² equals 0.017 m².

Key conversions include:
- 1 m² = 10,000 cm²
- 100 m² = 1,000,000 cm²
- 1 cm² = 0.0001 m²

To answer related questions:
- 1300 cm² = 0.13 m²
- 0.08216 m² is not directly related to the given information but can be converted to cm² by multiplying by 10,000.
- 170 cm² = 0.017 m²
- 60 cm² = 0.006 m²
- 12 m² = 120,000 cm²

Understanding this relationship makes it easier to convert between square meters and square centimeters.


**Modular Arithmetic**
Modular arithmetic is a system of arithmetic for integers where numbers 'wrap around' after reaching a certain value, known as the **modulus**. For example, in modulo 12 arithmetic, 8 + 6 equals 2, as 2 is the remainder after dividing 14 by 12. This operation is written as *a* % *n* in some programming languages.

**Definition of Modulo**
The 'mod' function can be implemented using the floor function: floor(z), where z is the greatest integer less than or equal to z. Both definitions allow *x* and *y* to be typed as integers or rational numbers.

**Applications of Modular Arithmetic**
Modular arithmetic has applications in number theory, abstract algebra, cryptography, and visual and musical art. It was first systematically studied by Carl Friedrich Gauss in the 18th century. Computers use modular arithmetic, where the modulus is 2^b (*b* being the number of bits of the values being operated on). For example, arithmetic operations on "int" integers are taken modulo 2^32 on most computers. In music, modular arithmetic is used to consider the twelve-tone equally tempered scale, especially in twelve-tone music. In visual art, modular arithmetic can be used to create artistic patterns based on multiplication and addition tables modulo *n*.

**Congruent Modulo**
Two integers *a* and *b* are **congruent modulo n**, written as *a* ≡ *b* (**mod** *n*), if their difference *a* - *b* is divisible by *n*, i.e., *a* - *b* = *kn* for some integer *k*. This idea is developed in the context of ring theory. The set of all equivalence classes is denoted **Z**/*n** **Z** = { [0]_n, [1]_n, [2]_n, ..., [*n*-1]_n}. If *a* and *b* are integers, the congruence *ax* ≡ *b* (**mod** *n*) has a solution if and only if the greatest common divisor (*a*, *n*) divides *b*.

**Properties of Congruence**
The equivalence relation has important properties: if *a*_1 ≡ *b*_1 (**mod** *n*) and *a*_2 ≡ *b*_2 (**mod** *n*), then *a*_1 + *a*_2 ≡ *b*_1 + *b*_2 (**mod** *n*) and *a*_1 * *a*_2 ≡ *b*_1 * *b*_2 (**mod** *n*). The set **Z**/*n** **Z** becomes a commutative ring with *n* elements, where addition and multiplication are well-defined operations.

**Rings and Fields**
In a commutative ring *R*, two elements *a* and *b* are **congruent modulo** an ideal *I* if *a* - *b* is an element of *I*. This turns out to be an equivalence relation, and addition and multiplication become well-defined operations on the factor ring *R*/*I*. The ring of integers **Z**/*n** **Z** is a field if and only if *n* is prime. Every finite field is an extension of **Z**/*p** **Z** for some prime *p*.

**Fermat's Little Theorem and Euler's Theorem**
If *p* is a prime number and *a* is any integer, then *a*^(*p*-1) ≡ 1 (**mod** *p*). For any integer *a* relatively prime to *n*, *a*^(φ(*n*)) ≡ 1 (**mod** *n*), where φ(*n*) denotes Euler's φ function counting the integers between 1 and *n* that are coprime to *n*. Euler's theorem is a consequence of the Theorem of Lagrange, applied to the group of units of the ring **Z**/*n** **Z**.


When creating a volcano plot using RankProd outputs, genes with a p-value of 0 pose a challenge for plotting -log10(p-value). These p-values are not the result of an error, but rather an underflow issue. The p-values are calculated as the number of permutations more extreme than the observed statistic divided by the total number of permutations. If no permuted statistics are as or more extreme than the observed statistic, the result is 0 divided by the number of permutations, yielding a p-value of 0.

An alternative approach is to calculate the p-value as the number of permutations more extreme than the observed statistic plus 1, divided by the total number of permutations. This method, employed by the limma package, counts the observed result as one of the permuted observations. For example, with 1000 permutations, this approach yields a p-value of 0.001 instead of 0. Although neither value is likely to be correct, 0.001 is more interpretable than a p-value of 0.

A related paper by B. Phipson and G. Smyth (Stat Appl Genet Mol Biol. 2010) supports this approach, stating that permutation p-values should never be zero. The paper discusses the calculation of exact p-values when permutations are randomly drawn, providing a basis for addressing underflow issues in p-value calculations.


**Mathematical Challenges**

1. **Number Patterns**: Design a new shape for 28 squares and arrange numbers in a logical way. Investigate patterns and relationships between numbers.

2. **Card Exchange**: If four children send one card to each of the other three, how many cards are sent in total? What if there were five or six children?

3. **Number System**: Explore what happens if all nines are removed from the number system. Rewrite numbers and examine the multiplications table.

4. **Jug Puzzle**: Use four jugs (9L, 7L, 4L, and 2L) to measure exactly 3L in three jugs.

5. **Jelly Feast**: Winifred Wytsh has four types of jelly. In how many ways can she create a feast with 32 legs?

6. **Plus Sign Puzzle**: Insert plus signs into the equation 1 2 3 4 5 6 7 8 9 to equal 99. How many solutions exist?

7. **Magic Square Maze**: Navigate a magic square maze, visiting every cell and exiting 15 times.

8. **Abundant Numbers**: Identify abundant numbers, which are less than the sum of their factors (excluding themselves).

9. **Page Numbers**: A book has 195 digits in its page numbers. How many pages does the book have?

10. **Pied Piper**: Investigate the number of people and rats in Hamelin, given the total number of legs.

11. **Dinner Party**: Seat 44 people at 15 square tables with 4 seats each, leaving no empty places.

12. **Spectacle Lenses**: Zumf makes lenses for residents of Zargon, who have 3 or 4 eyes. How many lenses are needed for 9 families?

13. **Finger Counting**: Count fingers in a series of pictures.

14. **Jigsaw Puzzle**: Complete two jigsaws, then combine them. Analyze the results of adding "touching" numbers and changing the jigsaws' positions.

15. **Caterpillar Shapes**: Arrange 16 caterpillar parts in a 4x4 square to form different shapes.

16. **Rabbit Pens**: Determine the number of each type of rabbit in a series of pens.

17. **Prison Cell Block**: Arrange 78 prisoners in a 12-cell block with 25 prisoners along each wall.

18. **Polo Square**: Place numbers 1-9 in a Polo Square so that each side adds to the same total.

19. **Dart Boards**: Explore combinations of scores on dart boards.

20. **Number Patterns**: Identify patterns in a series of numbers and create a new square with the same properties using numbers 1-16.

21. **Scoring 44**: With 5 darts, find different ways to score 44.

22. **Egg Timers**: Measure whole minutes using two egg timers (4 minutes and 7 minutes).

23. **Crossword Puzzle**: Arrange 5 different digits (0-9) in a cross, following specific rules.

24. **Digit Sum**: Calculate the sum of all 903 digits in a large number.

25. **Magic Square**: Change the magic constant of a magic square.

26. **Friend Ages**: Determine the ages of four friends (Cherri, Saxon, Mel, and Paul) using given information.

27. **Measuring Cylinders**: Relabel measuring cylinders after their labels are lost.

28. **Stamp Prices**: Determine the stamps needed to post a parcel weighing 825g, given prices for 1st and 2nd class mail.

29. **House Numbers**: Investigate the results of adding house numbers in different ways.

30. **Napier's Bones**: Use Napier's bones to calculate multiplication and division problems.

31. **Asteroid Game**: Play a game using addition, subtraction, multiplication, and division to blast asteroids.

32. **Function Machine**: Determine the output of a function machine given specific inputs (5, 7, and 4).

33. **Money Maze**: Navigate a maze, collecting and losing money, to find the highest and lowest returns.

34. **Crisp and Ice-Cream Prices**: Use logical thinking to determine the total cost of crisps and ice cream.

35. **Clock Face**: Draw a line on a clock face so that the numbers on both sides have the same total.

36. **Number Shapes**: Write numbers up to 64 in an interesting way to form a unique shape.

37. **Beanbag Hoops**: Place eight beanbags in hoops with specific requirements (4 in blue, 5 in red, and 6 in yellow).

38. **Planet Vuv**: Determine the number of Zios (3 legs) and Zepts (7 legs) on the planet Vuv, given a total of 52 legs.

39. **Triangle Puzzle**: Place digits 1-9 in circles so that each side of a triangle adds to the same total.

40. **Shape Combinations**: Find the totals of combinations of shapes with given values.

41. **Coin Distribution**: Distribute ten coins into three puddings, ensuring each pudding has at least two coins.

42. **Ring Scoring**: Score 100 by throwing rings on a board. Is there more than one way to achieve this?

43. **Sweet Shop**: Determine what each child can buy with their money, given prices for various sweets.

44. **Train Carriages**: Arrange 24 carriages into new trains. Find all possible combinations.

45. **Number Patterns**: Explore number patterns created using simple rules.

46. **Rod Measurements**: Measure lengths from 1 to 10 units using three rods of integer lengths (none longer than 10 units).

47. **Calendar Section**: Put a square box around specific dates in a calendar section. Add pairs of numbers and analyze the results.

**Multiple Choice Questions**

1. If four children send one card to each of the other three, how many cards are sent in total?
a) 6
b) 8
c) 12
d) 16

2. What is the sum of all 903 digits in a large number?
a) 903
b) 1000
c) 2000
d) Cannot be determined

3. How many lenses are needed for 9 families on the planet Zargon?
a) 27
b) 36
c) 45
d) 54

4. What is the magic constant of a magic square?
a) 15
b) 20
c) 25
d) 30

5. How many ways can you score 44 with 5 darts?
a) 1
b) 2
c) 3
d) 4

**Answers**

1. c) 12
2. d) Cannot be determined
3. c) 45
4. a) 15
5. c) 3


# Proving The Existence of Limits Examples 3

We will now look at some more examples of proving the existence of limits using the definition of a limit, that is $\lim_{x \to a} f(x) = L$ says that $\forall \epsilon > 0 \: \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \mid f(x) - L \mid < \epsilon$.

## Example 1

**Prove that $\lim_{x \to 2} \frac{x^2 + x - 6}{x - 2} = 5$.**

Let $\epsilon > 0$ be given and find $\delta > 0$ such that if $0 < \mid x - 2 \mid < \delta$ then $\biggr \rvert \frac{x^2 + x - 6}{x - 2} - 5 \biggr \rvert < \epsilon$.
Choose $\delta = \epsilon$. If $0 < \mid x - 2 \mid < \delta = \epsilon$, then $\biggr \rvert \frac{x^2 + x - 6}{x - 2} - 5 \biggr \rvert < \epsilon$.
Therefore $\lim_{x \to 2} \frac{x^2 + x - 6}{x - 2} = 5$.

## Example 2

**Prove that $\lim_{x \to a} \cos x = \cos a$.**

First let $\epsilon > 0$ be given. We want to find $\delta > 0$ such that if $0 < \mid x - a \mid < \delta$ then $\mid \cos x - \cos a \mid < \epsilon$.
By the Mean Value Theorem, $\exists C$ such that $A < C < B$ that satisfies $\mid \cos x - \cos a \mid = \mid -\sin C \mid \cdot \mid x - a \mid$.
Since $\mid -\sin C \mid = \mid \sin C \mid ≤ 1$, we have $\mid \cos x - \cos a \mid ≤ \mid x - a \mid$.
Choose $\delta = \epsilon$. Then $\mid \cos x - \cos a \mid ≤ \mid x - a \mid < \epsilon$ if $0 < \mid x - a \mid < \delta$.

## Example 3

**Prove that $\lim_{x \to 2} (x^2 - 4x + 5) = 1$.**

Let $\epsilon > 0$ be given and find $\delta > 0$ such that if $0 < \mid x - 2 \mid < \delta$ then $\mid (x^2 - 4x +5) - 1 \mid < \epsilon$.
We start with the inequality $\mid (x^2 - 4x +5) - 1 \mid = \mid (x - 2)(x + 2) \mid = \mid x + 2 \mid \cdot \mid x - 2 \mid$.
If we restrict $x$ to lie in some interval centered at $2$, we can assume $x$ is within a distance of $1$ from $2$.
Then $1 < x < 3 \implies 3 < x + 2 < 5 \implies \mid x + 2 \mid < 5$.
Therefore, $\mid x + 2 \mid \mid x - 2 \mid < 5 \mid x - 2 \mid < \epsilon \implies \mid x - 2 \mid < \frac{\epsilon}{5}$.
Choose $\delta = \mathrm{min} \{ 1, \frac{\epsilon}{5} \}$.
If $0 < \mid x - 2 \mid < \delta$, then $\mid (x^2 - 4x +5) - 1 \mid < \epsilon$.
Therefore $\lim_{x \to 2} (x^2 - 4x + 5) = 1$.


## Relative Reference

Excel uses relative references by default. In the formula in cell D2, the references to cells B2 and C2 are relative. When the formula is dragged down to cell D5, the references update to the neighboring cells. Specifically, cell D3 references cells B3 and C3, cell D4 references cells B4 and C4, and cell D5 references cells B5 and C5. This means each cell references its two neighbors on the left.

## Absolute Reference

In the formula in cell E3, an absolute reference to cell H3 is created by placing a $ symbol in front of the column letter and row number ($H$3). This allows the formula to be dragged down and across without changing the reference to cell H3. As a result, the correct lengths and widths in inches are calculated.

## Mixed Reference

A mixed reference combines relative and absolute references. In the formula in cell F2, a mixed reference is needed. To achieve this, a $ symbol is placed in front of the column letter of cell B2 ($B2) to fix the reference to column B. Similarly, a $ symbol is placed in front of the row number of cell B6 (B$6) to fix the reference to row 6. This allows the formula to be dragged down and across while maintaining the fixed references. The reference to the price is fixed to column B, and the reference to the reduction is fixed to row 6. The resulting formula can be quickly dragged to other cells, with the references to column B and row 6 remaining fixed.


To create a solenoid strong enough to levitate at the poles of the Earth's magnetic field, we can use Lenz's Law as an approximation. Assuming a 1kg, 100m long solenoid coil in the uniform Earth magnetic field, we need to calculate the required current.

The force required to levitate the solenoid is 9.8 Newtons, given by the equation:
F = 9.8 N
L = 100 meters
B = 50 uTesla

Using the equation Force = Current * Length x B-Field, we can calculate the required current:
(9.8 N) = I * (100 m) * (0.00005 T)
I = 1960 Amperes

However, this is impossible with copper wire due to its high resistivity. Using plasma as a conductor, we can estimate the resistive losses in the coil. Applying Spitzer Resistivity, we get:
n = 5.15 * 10^(-5) * (0.993 * 10) / ((1000)^(3/2))
n = 1.617 * 10^-8 ohm/m

Using Ohm's Law, we can calculate the coil's losses:
P = I^2 * R
P = (1960 A)^2 * (1.617 * 10^-8 ohm/m) * (100 m)
P = 6.212 W

The required plasma can be estimated by assuming a unit charge per molecule. A current of 1960 Amperes corresponds to approximately 1.55 * 10^22 charges per second. Using Avogadro's number, we can calculate the number of moles:
(1960 A) * (6.25 x 10^18 charges/coulomb) = (1.55 * 10^22 charges/s)
1.55 * 10^22 / Avagadro's Number = 0.02034 mols

Assuming the plasma is Xenon with a molar mass of 131 grams, we find that approximately 2.6 grams of Xenon are required.

However, as Scientist Smith YT points out, the calculations do not take into account other losses, such as voltage and stability. Using real-life measurements, Scientist Smith YT estimates that to lift off the ground with stability, the required energy would be approximately:
Amps: 73,039,904.53
Volts: 100,143,275.50
Watts: 7.31445549E5

This would require approximately 35.5 power plants to generate. The estimated magnetic flux would be:
Gauss: 740,137,656.60
Tesla: 74,013.76566

**Questions for the reader:**

1. What other losses are not taken into account in the initial calculation?
2. What is the power needed to maintain the 1000eV plasma temperature in the 100 meter tube?
3. What would the mass of a 100 meter glass tube be, to maintain the structural integrity needed to contain the plasma?

Note: The calculations and estimates provided are rough and may be far off from real values, as they are based on simplified models and assumptions.


# Citoequations 3

## Introduction to Citoequations

Citoequations are mathematical representations of concepts related to conscious survival and immortality. They are divided into several categories, including simple citoequations, advanced citoequations, and combined equations.

## Equations of Citotime

Citotime refers to the concept of time in relation to consciousness and ixperiencitness. Equations of citotime deal with the idea that consciousness and ixperiencitness can exist at different times, but not necessarily in synchronization.

## Oritiminterchangeability

Oritiminterchangeability refers to the concept that an original and its cidentireplica can exist at different times, but still produce the same behavior, consciousness, and ixperiencitness. The equation states that even if an original exists at a later time, it will produce the same behavior, consciousness, and ixperiencitness, but not at the same time.

## Orifuturistic Equations

Orifuturistic equations represent the concept of orifuturality, which states that if an original has the same structure and functioning at any time in the future, it will produce the same consciousness and ixperiencitness. The equation uses variables "i" and "j" to represent time, where i < j.

## Oricitofuturistic Equations

Oricitofuturistic equations represent the concept of oricitofuturality, which states that an original's current consciousness and ixperiencitness can exist at any time in the future, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is the present time and j is a future time.

## Citofuturistic Equations

Citofuturistic equations represent the concept of citofuturality, which states that an original's current consciousness and ixperiencitness can exist at any time in the future, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is the present time and j is a future time.

## Orihistoric Equations

Orihistoric equations represent the concept of orihistorality, which states that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is the present time and j is a past time.

## Oricitohistoric Equations

Oricitohistoric equations represent the concept of oricitohistorality, which states that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is the present time and j is a past time.

## Citohistoric Equations

Citohistoric equations represent the concept of citohistorality, which states that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is the present time and j is a past time.

## Oriquondamic Equations

Oriquondamic equations represent the concept of oriquondamality, which states that a past original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the past and j is the present time.

## Oricitoquandamic Equations

Oricitoquandamic equations represent the concept of oricitoquondamality, which states that a past original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the past and j is the present time.

## Citoquondamic Equations

Citoquondamic equations represent the concept of citoquondamality, which states that a past version of an original can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the past and j is the present time.

## Oriprospective Equations

Oriprospective equations represent the concept of oriprospectality, which states that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the future and j is the present time.

## Oricitoprospective Equations

Oricitoprospective equations represent the concept of oricitoprospectality, which states that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the future and j is the present time.

## Citoprospective Equations

Citoprospective equations represent the concept of citoprospectality, which states that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica. The equation uses variables "i" and "j" to represent time, where i is any time in the future and j is the present time.

## Key Concepts

* Citotime: the concept of time in relation to consciousness and ixperiencitness
* Oritiminterchangeability: the concept that an original and its cidentireplica can exist at different times, but still produce the same behavior, consciousness, and ixperiencitness
* Orifuturality: the concept that an original's structure and functioning at any time in the future will produce the same consciousness and ixperiencitness
* Oricitofuturality: the concept that an original's current consciousness and ixperiencitness can exist at any time in the future, produced by a cidentireplica
* Citofuturality: the concept that an original's current consciousness and ixperiencitness can exist at any time in the future, produced by a cidentireplica
* Orihistorality: the concept that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica
* Oricitohistorality: the concept that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica
* Citohistorality: the concept that an original's current consciousness and ixperiencitness can exist at any time in the past, produced by a cidentireplica
* Oriquondamality: the concept that a past original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica
* Oricitoquondamality: the concept that a past original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica
* Citoquondamality: the concept that a past version of an original can exist in the present, produced by a cidentireplica
* Oriprospectality: the concept that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica
* Oricitoprospectality: the concept that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica
* Citoprospectality: the concept that a future original's consciousness and ixperiencitness can exist in the present, produced by a cidentireplica

## Symbols

* F: represents the present time
* F+: represents a time after the present
* F-: represents a time before the present
* µ: represents a time after F+
* i: represents any time in the past or future
* j: represents any time in the past or future
* O: represents the original
* C: represents the cidentireplica
* E: represents the matter
* D: represents the space
* S: represents the structure
* F: represents the functioning
* U: represents the physical laws
* a: represents the itobody
* b: represents the itobody

## Equations

1. Oritiminterchangeability: O_i = C_j
2. Orifuturality: O_i = O_j
3. Oricitofuturality: O_i = C_j
4. Citofuturality: O_i = C_j
5. Orihistorality: O_i = O_j
6. Oricitohistorality: O_i = C_j
7. Citohistorality: O_i = C_j
8. Oriquondamality: O_i = C_j
9. Oricitoquondamality: O_i = C_j
10. Citoquondamality: O_i = C_j
11. Oriprospectality: O_i = C_j
12. Oricitoprospectality: O_i = C_j
13. Citoprospectality: O_i = C_j

Note: The equations are represented in a simplified form, using variables and symbols to represent the concepts and relationships between them.


Alpha Calculation

Alpha measures an investment's performance relative to a benchmark, beyond what beta would predict. The benchmark is typically the S&P 500, but can be any investment assigned in the Benchmark Dialog. A benchmark must be assigned before alpha calculations can be made. Alpha is expressed as an annualized return percentage, reflecting excess performance over the benchmark that beta alone would not predict.

#### Calculation Details:

To calculate alpha, simple monthly returns are calculated over the specified comparison period. The simple monthly return is:

Return = (End_price + Dist_per_share - Start_price) / Start_price

Excess returns are then calculated based on these returns and the risk-free return specified in the Benchmark Dialog:

Excess Return = Return - Risk_Free_Return

Monthly excess returns for both the investment and benchmark are calculated. For a 5-year comparison period, there are 60 monthly excess returns for each. To calculate alpha, these 60 data points are plotted with benchmark excess returns on the X-axis and investment excess returns on the Y-axis. The Y-axis intercept of the best-fit line through these data points is the alpha.

The formula for beta is used in this calculation, where 'n' is the number of data points (60 in this example), 'x' are the benchmark returns, and 'y' are the investment returns. Current interpolation range preferences are used when calculating returns. If sufficient pricing data is not available, alpha will not be reported.

The alpha for a portfolio, asset type, goal, or investment type is determined by calculating excess returns from a weighted average of the investments in that group, based on the ending value.

#### Advanced:

Fund Manager can output calculation data points to a log file. To enable logging, create a registry string value at HKEY_CURRENT_USER\Software\Beiley Software\Fund Manager\CurrentVersion\techanal, named "benchmarklog" with a value of "1". The log file, "stats_log.txt", is located in the user's AppData\Roaming\Fund Manager folder and is deleted upon exiting Fund Manager.

### See Also

* Benchmark Dialog
* Custom Report
* Beta Calculation
* Correlation
* Sharpe Ratio
* R-Squared
* Standard Deviation of Returns


**Mathematics Problems**

1. **Comparison**: 1/10 is greater than ??
Answer: 2

2. **Stress Element**: Draw and label the stress element for the given state of stress.
- a. xx = –500 psi, yy = 300 psi, zz = 0 psi, xy = 200 psi, yz = 0 psi, zx = 0 psi
- b. xx = 500 psi, yy = 400 psi, zz = –300 psi, xy = 200 psi, yz = 500 psi, zx = 0 psi
- c. xx = 500 psi, yy = 0 psi, zz = –300 psi, xy = 500 psi, yz = 0 psi, zx = 200 psi
Answer: 1

3. **Geometry**: From the figure and statement provided, select the proper statement to prove that through a point outside a line, one line can be drawn parallel to the line.
Answer: 1

4. **Simplification**: Simplify 4√2 + 7√2 - 3√2.
Answer: 8√2

5. **Prediction**: Use the relationship to predict the cost of a 20oz bag of coffee based on the given observations:
- 6oz is $2.10
- 8oz is $2.80
- 16oz is $5.60
Answer: 3

6. **Commission**: As a new employee, I earn an 8% commission.
No specific question provided.

7. **Volume of a Sphere**: What is the volume of a sphere with a radius of 6in? Round to the nearest hundredth.
Answer: 457.86 inches cubed or 394.56 inches cubed

8. **Unique Ways of Life**: Not a mathematics problem.

9. **Experiment**: What should you check if the experiment does not lead to the expected outcome?
No specific answer provided.

10. **Trigonometry**: Prove (csc^2x)(tanx) = 2csc^2x.
No proof provided.

11. **Quiz Scores**: Compare Lolo's and Ami's quiz scores.
No specific question provided.

12. **Photo Dimensions**: If an original photo is 5 inches by 8 inches, what are the dimensions of a new photo that is 4/8 the size?
No specific answer provided.

13. **Survey Results**: Analyze the survey results of 75 randomly selected students.
No specific question provided.

14. **Circle Properties**: Compare the properties of Circle A and Circle B.
No specific question provided.

15. **Function**: Evaluate the function h = 40t - 5t^2 when t is 1 and when t is 4.
No specific answer provided.

16. **Golf Ball Trajectory**: The path of a golf ball is represented by the function h = 40t - 5t^2.
No specific question provided.

17. **Money**: Sara has $84 more than Joline. Together they have $458. How much money does each person have?
No specific answer provided.

18. **Cannonball**: If a 20g cannonball is shot from a 5kg cannon with a velocity of 100m/s, then the speed of recoil of the cannon is -0.4m/s. True or False?
No specific answer provided.

19. **Pardon Power**: Who has the power to pardon individuals of crimes?
Not a mathematics problem.

20. **Jam Production**: Adriana jars 5 liters of jam every day. Write an equation that shows the relationship between the days x and the jam made y.
No specific equation provided.


### Hypergeometric Distribution

The hypergeometric distribution is a probability distribution that describes the number of successes in a fixed number of draws from a finite population without replacement. It is used when we want to find the probability of getting a certain number of successes in a sample drawn from a population.

#### Properties of Hypergeometric Distribution

The hypergeometric distribution has the following properties:

* The population size is finite, denoted by N.
* The sample size is fixed, denoted by n.
* The number of successes in the population is denoted by K.
* The probability of success is denoted by p = K/N.
* The probability of failure is denoted by q = (N-K)/N.

#### Example Problems for Hypergeometric Probability Distribution

1. Suppose we have a population of 100 items, 40 of which are successes. If we draw a sample of 10 items without replacement, what is the probability that we get exactly 3 successes?
2. A company has 50 employees, 20 of whom are women. If 10 employees are selected at random, what is the probability that exactly 4 of them are women?

#### Answers

1. Using the hypergeometric distribution formula, we get:
P(X = 3) = (40C3 \* 60C7) / 100C10 = 0.155
2. Using the hypergeometric distribution formula, we get:
P(X = 4) = (20C4 \* 30C6) / 50C10 = 0.238

#### Uses of Hypergeometric Distribution

The hypergeometric distribution is used in a variety of fields, including:

* Quality control: to determine the probability of defective items in a sample.
* Medicine: to determine the probability of a certain disease in a population.
* Social sciences: to determine the probability of a certain characteristic in a population.

#### Geometric Distribution

The geometric distribution is a probability distribution that describes the number of trials until the first success. It is used when we want to find the probability of getting the first success in a sequence of independent trials.

#### Example Problems for Geometric Distribution

1. Suppose we have a coin that has a probability of heads of 0.4. If we flip the coin repeatedly, what is the probability that we get the first head on the 3rd flip?
2. A company has a probability of success of 0.2. If they try repeatedly, what is the probability that they get the first success on the 5th try?

#### Answers

1. Using the geometric distribution formula, we get:
P(X = 3) = (0.6)^2 \* 0.4 = 0.144
2. Using the geometric distribution formula, we get:
P(X = 5) = (0.8)^4 \* 0.2 = 0.082

Note: The geometric distribution is a special case of the negative binomial distribution, where the number of successes is 1.


The total number of joint outcomes for rolling two dice is 6 times 6, which equals 36. This set of all possible outcomes is called the sample space of the probability experiment.

To write a sample space for dice, one method is to list all possible combinations of the two dice, resulting in a sample space of {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}. Alternatively, the sample space can be represented by adding the numbers on the two dice, giving {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

The number of elements in the sample space for rolling two dice is 36. Since each die has 6 possible outcomes, there are 6 * 6 = 36 total combinations possible from rolling two dice.

When a single die is rolled, there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6. The probability of rolling a specific number with a six-sided die can be calculated using a probability table. For example, the probability of rolling a 4 is 3/36, or approximately 8.333%.

It's worth noting that "die" is the singular form of "dice," so "roll a die" means rolling one, while "roll the dice" means rolling two. In the context of probability, a sample space can have any number of elements, depending on the experiment. For a single coin toss, the sample space has 2 elements, while for a single die roll, the sample space consists of 6 elements: {1, 2, 3, 4, 5, 6}.

A six-sided die is said to be unbiased if it is equally likely to show any of its six sides. When an unbiased die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6}, with a total of 6 possible outcomes.

Multiple choice questions:
1. How many elements are in the sample space for rolling two dice?
A) 12
B) 24
C) 36
D) 48

Answer: C) 36

2. What is the probability of rolling a 4 with a six-sided die?
A) 1/36
B) 3/36
C) 6/36
D) 8/36

Answer: B) 3/36

3. How many possible outcomes are there when rolling a single die?
A) 4
B) 5
C) 6
D) 8

Answer: C) 6


# Systems of Two Simultaneous Linear Equations in Two Unknowns

Systems of two simultaneous linear equations in two unknowns have the shape:
\[ax + by = c\]
\[dx + ey = f\]
where \(a, b, c, d, e, f\) are numerical coefficients, and \(x, y\) are unknowns.

## Basic Methods of Solution

There are two basic methods to solve these simultaneous equations: Substitution and Addition or Subtraction.

### Substitution

1. Express one unknown in terms of the other using one equation.
2. Substitute this expression into the second equation.
3. Solve for the other unknown.
4. Substitute the value of the second unknown back into the expression for the first unknown.

Example: Solve the system of simultaneous equations:
\[3x + 2y = 8\]
\[x + 3y = 5\]

From the first equation, express \(x\) in terms of \(y\):
\[x = (8 - 2y) / 3\]

Substitute this expression into the second equation:
\[(8 - 2y) / 3 + 3y = 5\]
Solving for \(y\), we get:
\[y = 1\]

Now, substitute \(y = 1\) back into the expression for \(x\):
\[x = (8 - 2*1) / 3 = 2\]

### Addition or Subtraction

1. Multiply both sides of the first equation by a suitable constant to eliminate one variable when added to or subtracted from the second equation.
2. Add or subtract the two equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the value of the variable back into one of the original equations to solve for the other variable.

Example: Solve the system of simultaneous equations:
\[x + 2y = 4\]
\[3x + 6y = 12\]

Multiply the first equation by -3 and add it to the second equation:
\[-3x - 6y = -12\]
\[3x + 6y = 12\]
This results in \(0 = 0\), which means the equations are dependent and have an infinite number of solutions.

## The Second Order Determinants

The solution to a system of two simultaneous linear equations can be expressed using second-order determinants:
\[x = \frac{\begin{vmatrix} c & b \\ f & e \end{vmatrix}}{\begin{vmatrix} a & b \\ d & e \end{vmatrix}} = \frac{ce - bf}{ae - bd}\]
\[y = \frac{\begin{vmatrix} a & c \\ d & f \end{vmatrix}}{\begin{vmatrix} a & b \\ d & e \end{vmatrix}} = \frac{af - cd}{ae - bd}\]

## Cramer’s Rule

Cramer’s rule states that the solution to a system of two simultaneous linear equations can be found using the formulas:
\[x = \frac{\begin{vmatrix} c & b \\ f & e \end{vmatrix}}{\begin{vmatrix} a & b \\ d & e \end{vmatrix}}\]
\[y = \frac{\begin{vmatrix} a & c \\ d & f \end{vmatrix}}{\begin{vmatrix} a & b \\ d & e \end{vmatrix}}\]

Example: Solve the system of simultaneous equations:
\[x + y = 12\]
\[2x - 3y = 14\]

Using Cramer’s rule:
\[x = \frac{\begin{vmatrix} 12 & 1 \\ 14 & -3 \end{vmatrix}}{\begin{vmatrix} 1 & 1 \\ 2 & -3 \end{vmatrix}} = \frac{12*(-3) - 14*1}{1*(-3) - 2*1} = \frac{-36 - 14}{-3 - 2} = \frac{-50}{-5} = 10\]
\[y = \frac{\begin{vmatrix} 1 & 12 \\ 2 & 14 \end{vmatrix}}{\begin{vmatrix} 1 & 1 \\ 2 & -3 \end{vmatrix}} = \frac{1*14 - 12*2}{1*(-3) - 2*1} = \frac{14 - 24}{-3 - 2} = \frac{-10}{-5} = 2\]

## Investigation of Solutions

The investigation of solutions to a system of two simultaneous linear equations in two unknowns shows that there are three possible cases:

1. **Single Solution**: If the coefficients of the unknowns are disproportionate (\(a:d \neq b:e\)), the system has a single solution.
2. **Infinite Solutions**: If all coefficients are proportional (\(a:d = b:e = c:f\)), the system has an infinite number of solutions.
3. **No Solution**: If the coefficients of the unknowns are proportional but not to the free terms (\(a:d = b:e \neq c:f\)), the system has no solution.

Example: Determine the number of solutions for the system:
\[2x - 3y = 7\]
\[4x - 6y = 12\]

Dividing the first equation by 2 and the second by 4 gives:
\[x - \frac{3}{2}y = \frac{7}{2}\]
\[x - \frac{3}{2}y = 3\]

These equations are contradictory, as the same expression \(x - \frac{3}{2}y\) cannot be equal to both \(\frac{7}{2}\) and 3 simultaneously. Therefore, the system has no solution.


The least squares method provides the closest relationship between the dependent and independent variables by minimizing the distance between the residuals, and the line of best fit, i.e., the sum of squares of residuals is minimal under this approach. Hence the term “least squares.” Examples of least squares Regression Line example.

For example, Master Chemicals produces bottles of a cleaning lubricant. Now that we have determined the loss function, the only thing left to do is minimize it. The least squares problem, in our case A0A will always have a solution, even if it is singular.)

We can solve rf(x) = 0 or, equivalently AT Ax = AT b to nd the least squares solution. The fundamental equation is still A TAbx DA b.

To test Least Squares Regression Line example, suppose we wanted to estimate a score for someone who had spent exactly 2.3 hours on an essay.

First, we want to solve an example of a rst order ordinary di erential equation as an illustration to present least squares methods. An example of using LSM to solve a rst-order ODE 1.1.1.

A Quiz Score Prediction Fred scores 1, 2, and 2 on his first three quizzes.

The least squares solution).


**Value Betting: A Strategy for Generating Stable Winnings**

Value betting is a complex strategy that offers advantages over surebetting, but also comes with more risk. It gives the player an edge over the bookmaker, allowing for stable winnings. To understand value betting, let's examine the bookmaking process.

**The Bookmaking "Kitchen"**

Imagine a coin flip with two possible outcomes: heads or tails. The probability of each outcome is 50%. A fair odds setup would be 2 for heads and 2 for tails. However, bookmakers use odds like 1.9 for heads and 1.9 for tails, ensuring a commission, known as the **margin**, is paid to them. This margin is calculated as `1/K1 + 1/K2 + … + 1/Kn = 1`, where `K1`, `K2`, … `Kn` are the odds for each outcome.

In reality, bookmakers set odds to give themselves an advantage. Using Bwin's odds for a Chelsea – Manchester United match as an example: `1/1.95 + 1/3.3 + 1/3.75 = 1.0825`, resulting in an 8.25% margin. This means bookmakers always have an edge over players.

**Value Betting: Identifying Mistakes in Bookmaker Odds**

Bookmakers sometimes make mistakes, setting high odds for an outcome. If a player identifies such a mistake, they can place a **valuebet**, which is a bet where `K * P > 1`, with `K` being the odds and `P` being the real probability of the outcome. For example, if a bookmaker sets odds of 2.1 for heads and 1.75 for tails in a coin flip, a player can exploit this mistake by betting on heads.

To find valuebets, a player must assess the real probability of event outcomes. There are two methods: **analytical** and **statistical**. The analytical method involves researching teams' potentials, statistics, and other factors, while the statistical method uses odds from multiple bookmakers to calculate the bookmaker's margin and determine the probability.

**Finding Valuebets**

Our service uses the statistical approach to find valuebets. By analyzing odds from multiple bookmakers, we can calculate the margin and determine the probability of each outcome. This method can be automated and provides an accurate prediction of the outcome.

**Surebets and Valuebets**

A bookmaker's surebet always has a negative margin when calculated as if all bets were placed by a single bookmaker. This means that at least one of the bets in the surebet is overestimated, creating a valuebet. By analyzing surebets, we can identify the overestimated bet and use it to our advantage.

**Advantages and Disadvantages of Valuebets**

Valuebets have several advantages over surebets:

* Fewer bets: With valuebets, you only need to place one bet per valuebet, minimizing the risk of not being able to place other bets.
* Fewer accounts: You only need one account with a bookmaker to play with valuebets.
* More natural behavior: With valuebets, you can bet any amount that doesn't contradict your strategy, raising less suspicion.
* Higher profitability: In theory, valuebetting is more profitable than surebetting, as you're not paying for insurance against losing.

However, valuebets also come with disadvantages:

* No winning guarantee: With valuebets, each bet can be lost, and there's no guarantee of winning.
* Risk mitigation: To mitigate this risk, it's essential to place many small bets instead of making a few large ones.

By understanding value betting and its advantages, players can develop a long-term strategy for generating stable winnings.


### Granular Entropy

The concept of entropy is crucial in thermodynamics, and its importance was realized during an experiment on the high-temperature reaction of metals forming alloys. The experiment involved constructing a high-temperature calorimeter, which consisted of a vacuum system, quartz glass ampoules, and a tantalum wire heating coil with a power supply of 50 amps at 115 volts. This setup allowed for the measurement of the heat of formation of a compound, also known as the enthalpy of formation.

Entropy plays a significant role in thermodynamics, as evident from the equation ΔG = ΔH - TΔS, where ΔG is the Gibbs free energy change, ΔH is the enthalpy of vaporization, and ΔS is the vaporization entropy. In the case of boiling, the Gibbs free energy change is zero, indicating that entropy is as important as enthalpy. The Boltzmann entropy formula, S = k \* ln(Ω), where k is the Boltzmann constant and Ω is the number of states accessible to a system, provides a way to calculate entropy.

The Boltzmann constant is 1.38062 x 10^(-23) joule/kelvin, and the number of states accessible to a system can be vast. For example, the number of ways to arrange 128 tennis balls in a fixed volume is approximately 10^250, which exceeds the total number of atoms in the universe, estimated to be 10^80. This concept of granular entropy has applications in understanding the behavior of materials such as soil, sand, and snow.

Researchers at the University of Cambridge have developed a technique to calculate the configurational entropy of a granular system, which is essential for predicting the behavior of these materials. The technique involves selecting a small sample of configurations and calculating the probability of their occurrence. This approach can be applied to other areas, including machine learning, where it is useful to know all the different ways to connect a system to process information.

The study of granular entropy has significant implications for understanding the behavior of materials and predicting phenomena such as avalanches. The research team's findings suggest that the technique can be used to estimate the total configurational entropy of a system, which is essential for understanding the behavior of granular materials. The technique's applicability to other areas, such as machine learning, makes it a valuable tool for researchers.

In conclusion, the concept of granular entropy is crucial in understanding the behavior of materials and predicting phenomena such as avalanches. The Boltzmann entropy formula provides a way to calculate entropy, and the technique developed by researchers at the University of Cambridge offers a valuable tool for estimating the total configurational entropy of a system. The applications of granular entropy extend beyond materials science to areas such as machine learning, making it a vital area of research.

References:

1. Stefano Martiniani, K. Julian Schrenk, Jacob D. Stevenson, David J. Wales, and Daan Frenkel, "Turning intractable counting into sampling: Computing the configurational entropy of three-dimensional jammed packings," Phys. Rev. E, vol. 93, no. 1 (January, 2016), Document no. 012906.
2. How many ways can you arrange 128 tennis balls? Researchers solve an apparently impossible problem, St. John's College, Cambridge, Press Release, January 27, 2016.
3. Ning Xu, Daan Frenkel, and Andrea J. Liu, "Direct Determination of the Size of Basins of Attraction of Jammed Solids," Phys. Rev. Lett., vol. 106, no. 24 (June 17, 2011), Document no. 245502.
4. Daniel Asenjo, Fabien Paillusson, and Daan Frenkel, "Numerical Calculation of Granular Entropy," Phys. Rev. Lett., vol. 112, no. 9 (March 7, 2014), Document no. 098002.


# From The Archives: Al-Khwarizmi Does The Math

The mathematician and astronomer Muhammad ibn Musa al-Khwarizmi (ca. 783-847) is the person behind the development of multiplication and division methods. His name is the origin of the term "algorithm".

Al-Khwarizmi's life is not well-documented, but it is believed that he was born in the region of Khwarazm in what is now Uzbekistan. He may have been a Zoroastrian who converted to Islam.

As a scholar in al-Mansur's court in Baghdad, al-Khwarizmi introduced "Hindu numerals" to the Muslim world. He created an important astronomical chart, known as a *zij*, which enabled calculations of the positions of the sun, moon, and major planets, and allowed for time-telling based on stellar and solar observations.

Al-Khwarizmi's most significant contribution to science is his mathematical treatise, *al-Kitab al-Mukhtasar fi Hisab al-Jebr wal-Muqabala*, also known as *al-jebr* or algebra. The treatise combines mathematical theory with practical examples related to inheritances, property division, land measurements, and canal digging. He is credited with inventing quadratic equations and word problems, including classics that were still used centuries later.

The next time you need to solve a problem like "how long it will take for two cars to meet in Dubuque if one car leaves Minneapolis going 60 miles an hour and the other leaves Peoria traveling 75 miles an hour?", remember to thank al-Khwarizmi.

Key facts about al-Khwarizmi include:
- Born: ca. 783
- Died: ca. 847
- Introduced Hindu numerals to the Muslim world
- Created an important astronomical chart
- Wrote the mathematical treatise *al-Kitab al-Mukhtasar fi Hisab al-Jebr wal-Muqabala*
- Invented quadratic equations and word problems

Note: The original text included a personal message and comments from readers, which have been removed to refine the content and focus on the key information about al-Khwarizmi.


The **CONFIDENCE.NORM** function in Excel calculates the half-width of a confidence interval for a population mean. The syntax is **CONFIDENCE.NORM(alpha, standard_dev, size)**, where **alpha** is the significance level, **standard_dev** is the population standard deviation, and **size** is the sample size.

For example, to calculate a 95% confidence interval for the mean height of a plant species, you can use the function if you know the population standard deviation and sample size. Suppose you collect a sample of **50** plants with a mean height of **14.5** inches, and the population standard deviation is **3.1** inches. You can use the formula =**CONFIDENCE.NORM(0.05, 3.1, 50)** to calculate the half-width of the 95% confidence interval, which is **0.85926**.

To construct the confidence interval, add and subtract this value from the sample mean: **14.5 ± 0.85926**, resulting in a 95% confidence interval of (**13.64074, 15.35926**). The confidence interval for the mean height is between **13.64074 inches** and **15.35926 inches**.

Note that a higher confidence level results in a wider confidence interval. For instance, a 99% confidence interval using =**CONFIDENCE.NORM(0.01, 3.1, 50)** yields a half-width of **1.12926**, resulting in a confidence interval of (**13.37074, 15.62926**), which is wider than the 95% confidence interval.

Key points to remember:

* The **CONFIDENCE.NORM** function calculates the half-width of a confidence interval for a population mean.
* The function uses the syntax **CONFIDENCE.NORM(alpha, standard_dev, size)**.
* A higher confidence level results in a wider confidence interval.
* To construct the confidence interval, add and subtract the half-width from the sample mean.

Additional resources are available for other common operations in Excel, including calculating confidence intervals for regression coefficients, plotting confidence intervals, and performing hypothesis testing.


**Advances in Pure Mathematics** Vol.4 No.4(2014), Article ID:44634,5 pages DOI:10.4236/apm.2014.44017

Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry

Yong Liu, Jun Wang

Department of Mathematics and Physics, North China Electric Power University, Beijing, China

Email: liuyong@ncepu.edu.cn, junwang3000@163.com

Copyright 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 20 February 2014; revised 20 March 2014; accepted 28 March 2014

ABSTRACT

The Allen-Cahn equation on the plane has a 6-end solution with regular triangle symmetry. We prove that this solution is nondegenerated in the class of functions possessing regular triangle symmetry. As an application, we show the existence of a family of solutions close to this solution.

**Keywords:** Allen-Cahn Equation, Multiple-End Solution, Nondegeneracy

1. Introduction

We consider the Allen-Cahn equation:

∆u + u(1 - u^2) = 0

A solution to this equation is called a multiple-end solution if outside a large ball, the nodal curves of u are asymptotic to finitely many half straight lines. The most simple example of a multiple-end solution is the saddle solution. For each k ≥ 3, the equation has a solution with regular polygon symmetry.

2. Main Result

Our main theorem is the following nondegeneracy result:

Theorem 1.1 Assume u is a solution of the equation and satisfies certain symmetry and asymptotic conditions. Then the linearized operator around u has a trivial kernel.

With the help of this theorem and the moduli space theory, we have the following Corollary 1.1:

Corollary 1.1 There is a family of 6-end solutions, different from the given solution, such that in the limit as the parameter goes to zero, the solutions converge to the given solution in a certain sense.

3. Proof of Theorem 1.1

To prove our main theorem, we use the ideas developed in previous work. We first show that the linearized operator around u has the same symmetry as the function u. Then we show that the nodal set of u is bounded. Finally, we use these results to prove that the linearized operator around u has a trivial kernel.

4. Proof of Corollary 1.1

By Theorem 1.1, the solution is nondegenerated in the class of functions having regular triangle symmetry. Then we can use the moduli space theory to show the existence of a family of 6-end solutions satisfying the desired properties.

5. Conclusion

In this paper, we prove the nondegeneracy of the 6-end solution to the Allen-Cahn equation with regular triangle symmetry. As an application, we show the existence of a family of solutions close to this solution. We conjecture that the angles of consecutive ends of these solutions should be exactly equal to π/3, but this seems to be a difficult problem.


The quadratic formula is a reliable method for solving the quadratic equation, which is expressed as ax^{2} + bx + c = 0. Although other methods, such as factoring, graphing, and completing the square, can be used to solve quadratic equations, the quadratic formula is the only method that always works. It allows users to solve for x directly by altering the quadratic equation.

The quadratic formula has key advantages over other methods, including simplicity and ease of use. Unlike completing the square, it does not require complex steps to be remembered. Users can simply plug in the coefficients a, b, and c and solve for x. The formula provides two solutions, one positive and one negative, which can be real or complex.

The quadratic equation has several real-world applications, making the quadratic formula a useful tool. For example, it can be used to model the trajectory of a thrown ball, taking into account the initial height, velocity, and gravitational pull. By using the quadratic formula, users can determine when the ball will land on the ground. The formula's versatility and reliability make it a valuable resource for solving quadratic equations in various contexts.


# Munching on Inscribed Angles

The Inscribed Angle Theorem states that an angle inscribed into a circle measures half the central angle subtending the same arc. This theorem has been used in various contexts, including Sine, Cosine, and Ptolemy's Theorem. A proof and dynamic illustration of this theorem are available elsewhere.

Proposition III.21 concludes that all inscribed angles in the same circle subtending the same arc are equal. This proposition can be turned around to ask: if ∠APB = ∠AQB, then does a circle exist that passes through points A, B, P, and Q? The answer is yes. To demonstrate this, inscribe ΔAPB into a circle. Point Q may lie inside, outside, or on the circumference of this circle. However, the cases where Q lies inside or outside the circle are impossible.

By Proposition I.16, we can consider a configuration where Q lies outside the circle. This leads to the question: in a triangle ABC, do points B, C, Hb, and Hc lie on the same circle?

The Inscribed Angle Theorem can be reformulated to state that the locus of all points P, such that ∠APB is equal to a given angle, consists of two circular arcs symmetric with respect to segment AB. These arcs merge into a circle when the angles are right. In this case, triangles APB are right triangles, and AB serves as the common hypotenuse and diameter of the circle.

Thales' Theorem states that an angle inscribed in a semicircle is right, and vice versa. This theorem can be used to show that the locus of the foot of the perpendicular from point B onto lines from a given pencil through A is a circle.

A pencil of lines can be visualized as a single line rotating around a fixed center. As the line rotates, the foot of the perpendiculars dropped onto the line from point A marks a circle. Similarly, the perpendicular onto the line from B rotates around B, creating a motion that can be analyzed.

The equality ∠APB = ∠AQB leads to the interpretation that if a line through point A rotates around A with a constant angular speed, and another line rotates around point B with the same angular speed, then the angle formed by the two rotating lines remains the same. This means that points P, Q, A, and B lie on the same circle.

Fixing points A, B, and P, and letting Q change, we observe that the point of intersection of two lines rotating with equal angular velocities describes a circle. This can be used to make an observation about cyclic quadrilaterals: in a convex quadrilateral ABPQ, the opposite sides add up to 180°, while in a self-intersecting quadrilateral APBQ, the opposite angles are equal.

### Angles in Circle

- Angle Subtended by a Diameter
- Inscribed Angles
- Inscribed and Central Angles in a Circle
- Munching on Inscribed Angles
- Sangaku with Angle between a Tangent and a Chord
- Secant Angles in a Circle
- Secant Angles in a Circle II
- Thales' theorem

Copyright © 1996-2018 Alexander Bogomolny71683039


# Draw All Lines Of Symmetry For The Following Figure

A shape is symmetrical if it has at least one line of symmetry, a line that divides the shape into two mirror-image halves. The line is called the axis of symmetry or line of symmetry. For a line to be a line of symmetry, every point on one side of the line must have a corresponding point on the other side that is equidistant from the line.

To draw lines of symmetry, identify the figures which have the same distance from the central part, that is, the figures which have the point of symmetry. Then, draw the lines that divide the shape into two equal parts. The lines may be vertical, horizontal, or diagonal lines.

## Types of Symmetry

* Line symmetry: A shape is symmetrical if it has at least one line of symmetry.
* Rotational symmetry: A shape is symmetrical if it looks the same after a rotation of 90, 180, 270, or 360 degrees.

## Drawing Lines of Symmetry

1. Trace each figure and draw the lines of symmetry, if any.
2. Identify the central part of the figure and draw lines that divide it into two equal parts.
3. Check if the figure has vertical, horizontal, or diagonal lines of symmetry.

## Examples

* An equilateral triangle (3 sides) has 3 lines of symmetry.
* A square (4 sides) has 4 lines of symmetry.
* A quadrilateral with a horizontal line of symmetry but no vertical line of symmetry has only 1 line of symmetry.

## Questions

1. What is a line of symmetry?
 Answer: A line that divides a shape into two mirror-image halves.
2. How many lines of symmetry does a human body have?
 Answer: 1 (vertical line of symmetry)
3. What is rotational symmetry?
 Answer: A shape is symmetrical if it looks the same after a rotation of 90, 180, 270, or 360 degrees.

## Exercises

1. Draw the net of a square pyramid with base as square of side 7 cm and slant edges 9 cm.
2. Trace each figure and draw the lines of symmetry, if any.
3. Identify the figures which have the same distance from the central part, that is, the figures which have the point of symmetry.