# Tagged Questions

Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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### Best strategy for the first player in a game for two on a large checkered paper

Here's a puzzle that's been seating in the back of my head for quite a long time. The game is played on a grid of infinite dimensions; sufficiently large checkered paper. First player specifies ...
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### The Jeep Problem with Equally Spaced Stations

Consider the following problem. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a ...
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### The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
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### Find the Wrong Student

There are 15 student in the class and each of them has a different number 1 to 15. Student #1: wrote the natural number on the board. Student #2 said : This number is divisible by my number(number ...
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### On a possibility/impossibility of a certain twisted situation in a tournament

Recently I encountered the following puzzle: Consider a game for two players which can only result in a win of one of the players (no ties). Now $n$ players decided to play this game each with ...
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### Knowing the rates of gain and loss, estimate the average amount.

This question was presented to me as such: Say that people grow 100 hairs a day and lose 100 hairs a day and hairs have a lifespan of 100 days. Estimate how much hair an average person would have. ...
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### Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
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### Probability of solution amongst a set of Diophantine equations

I have a set of Diophantine equations which I know only one equation has a single solution. I am trying to find a way to give probabilities to which equation contains the solution. For example, ...
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### Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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### Bribing a friend for money (game)

Here's how the game works: I get a pay bonus. I'm so happy I decide to give away $100$ dollars to one of my $n>1$ friends. My friends try to bribe me to get the money for themselves; I'm a ...
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### Is this a pure imaginary number?

I've met this formula and I need to demonstrate that it is purely imaginary (it has no real part). $\frac{1}{2}\log(-\exp(i2\pi q))$, //for a real "input" q. As I don't know much about maths, what I'...
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### enumerating polyominos

Polyominoes are made by gluing together finitely many squares along their edges. They always have connected interiors, but are allowed to have holes. Enumerating polyominoes is a huge subject, and ...
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### Weighted War - Game of Mind and Probability

Weighted War is a game of bidding, where: Both players have cards valued from $1$ to $11$ in their hands There is a third pile of cards from $1$ to $11$ face down on the table and shuffled,...
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### Make y the subject of x = y/(y-z)

I'm struggling with this GCSE question, but I think I'm just being silly. I've removed the fraction, making it: x(y-z) = y And then tried removing the brackets, making it: xy-xz = y But I'm not ...
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### How math help reduce terms and conditions of someone's dying wish?

Good morning everyone... This is my very first question here, so I apologise in advance for any wrongdoing which I possibly make unintentionally. So here is a little background story. I'm working at a ...
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### The Thirty-one Game: Winning Strategy for the First Player

I am going through UCLA's Game Theory, Part I. Below is an exercise on page 6: The Thirty-one Game. (Geoffrey Mott-Smith (1954)) From a deck of cards, take the Ace, 2,3,4,5, and 6 of each suit. ...
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### How can I interpret joint and conditional entropies?

Say I have three variables, X,Y, and Z. Joint Entropy of X and Y is greater than joint entropy of X and Z, what does it mean? and Similarly, conditional entropy of X and Y is greater than conditional ...
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### What's the best way to catch wild Pokémon in Pokémon GO?

In the newly released Pokémon GO, one of the major activities of the game is to catch wild Pokémon. These Pokémon are shown in the "nearby" list and their "rough distance" (RD) to you can be 0, 1, 2, ...
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### Sets of integers with few sums

Let $S$ be a finite set of integers. Denote by $S^{\leq k}$ the set $\{a_1+\dots+a_\ell : \ell\leq k, a_1,\dots,a_\ell\in S\}$ of sums of at most $k$ elements from $S$. What are best/worst cases for ...
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### Recurrence equation for sequence of vectors

Consider recurrent formula for a sequence of numbers $(y_n)$ (either real or complex): $$a_k y_{n+k}+a_{k-1}y_{n+k-1}+\cdots+a_0y_n=\sum_{i=0}^k a_i y_{n + i} = 0$$ It's known that the exact explicit ...
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### Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
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### Sum of slopes between points of concurrent normals to hyperbola is zero

Let $A_1, A_2, A_3, A_4$ be four points on the hyperbola $xy = 1$. Suppose that the normals to the hyperbola at these four points are concurrent, i.e. they intersect in a single point. Prove that ...
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### Maximum Coin Changes That Does Not Add To a Dollar

What is the maximal amount of money attained from coins of 1, 5, 10, 25 cent denominations that none of its subset amounts to 100 cents? We can find the solution with exhaustive or naive dynamic ...
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### Heegner Prime visualizations

The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ring of integers $\textbf{Q}(\sqrt{-d})$ have unique factorizations. 1 gives the Gaussian integers. 3 gives the Eisenstein integers. 7 ...
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### Bound on the value of $\binom{n }{n/2}$

I know the value of $\binom{n}{r}$ is maximum for $r=n/2$ if $n$ is even. I am in need to calculate the value of $\binom{n}{n/2}$. \begin{align*} \binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\ldots+\binom{...
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### Integer solution to the equation below

I wanted to know integer solutions to the equation (1/k1) + (10/k2) + (100/k3) + ..... + (10^18/k19) = 1 (where k1,k2,k3.... are integers) which I believe is ...
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### Relative speed of minute hand in a clock

In a 12 hours clock: The minute hand has to chase the hour hand with a relative speed of 5.5 degrees/min what is the mathematical derivation of this relative ...
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### Movie math, real or not? (Ghost busters trailer) [closed]

I was watching the ghost busters trailer, and I stopped on a frame containing 'quantum physics'. I wonder if all of this is meaningful math, or some of it is complete gibberish, to look smart. I see ...
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### Hard polynomial question.

I stuck with the following question. Let $n\geq 1$, $A, B\in\mathbb{R}[x]$ be polynomials of degree $n+1$ and $n$, respectively. Let $a_i$ and $b_i$ be its roots, suppose they are all real and ...
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### Is it possible to find an expression for the x-value of where the normal line to a function hits another function? (Picture shown)

I was dabbling in some recreational math and I stumbled upon this problem: Say there are two arbitrary functions $f(x)$ and $g(x)$. Create the normal line to $g(x)$ at $x_g$; now find an expression ...
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### Cryptogram Problem [closed]

Phil is thinking of a number less than 700. When he interchanges the ones and the hundreds digits, the number increases by 297. When he interchanges the ones and the tens digits, the result increases ...
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### Throwing eggs from flooors, general case [duplicate]

The following problem and its solution is well-known: you have two eggs and you have to measure which is the first floor of a building (that has in this example 36 floors) from which if an egg is ...
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### How can I understand the step by step calculations for the formula from the blog below?

I am studying clustering and found a useful article on the blog post here Finding the K in K-Means. But I am having difficulty in understanding the formulas below and how I can do step by step ...
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### Smallest future date which involves no repetition of a digit in the format DD/MM/YYYY [closed]

What is the smallest future date which involves no repetition of a digit in the format DD/MM/YYYY for the year? What is your approach?
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### Mean Square Displacement of Alternating Random Walk

Consider a 1D random walk with varying steps: the length of the steps is $A$ a fraction $\gamma$ of the time, and $B$ the rest of the time. If $\gamma = 0$, the mean squared displacement approaches ...
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### What's the upper bound for sofa problem?

I have seen a claim that for the sofa problem, an upper bound for the area of a sofa is $2 \sqrt 2$, and that this can be proved by a "simple" argument. But I can't find a proof. What that argument? (...
### Ways to squeeze $e$ by hand
Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where \$b-a&...