Given a chessboard with an arbitrarily-flipped coin on each square, can I communicate a specific square on the board to you by flipping a single coin on the board?

Compute the next value of a polynomial given that its first 1008 values are the first 1008 odd-indexed Fibonacci numbers

We prove a theorem with implications for conservation of mass, momentum and energy in kinetic theory.

Given that the squares of the sides and diagonals of a quadrilateral are all rational, prove that the ratio of areas of any two triangles formed by taking three out of the four vertices is also rational

Prove that a given recursively defined sequence is periodic

At long last, the Riddler problem from over two years ago. Freeze tag!

Transforming a problem involving the integer lattice in n-dimensions to a tessellation problem.

Determine all possible values of a polynomial expression in three variables.

Given n > 3 points on a circle, what is the chance that the convex polygon formed has an acute angle?

A difference equation pops up in a derivation of the Poisson Process, which reminds an old dog of some calculus he learned back in physics.

Six snails are situated at the corners of a regular hexagon whose sides are all 10 meters. Each snail is determined to reach its clockwise neighbor, and they all travel at the same speed. How long is each snail's slimy trail?

A short problem taken from the 2008 Putnam.

You have 4 coins sitting at the four corners of a table. They are set to heads or tails randomly, and you are blindfolded. Your goal is to try to get all the coins to be on heads, after which the game immediately ends. The catch is that after each round, the table rotates by 90, 180, 270, or 360 at random. How do you win this game?

Prove that a given complex function (parametrized with an integer n) has no zeros in the closed unit disk

Find all pairs of positive integers such that the sum of the reciprocals of the two integers equals 3/2018

Given N distinct red and N distinct blue points on a 2D plane, no three of which are colinear, prove that there exists a way to connect (using line segments) each red point to a unique blue point such that no two line segments intersect.

We'll prove the uncertainty principle using basic principles from probability and statistics.

A quick neat proof of Fermat's Little Theorem using basic modular arithmetic. Also, welcome to the new website!

A rope rests on two platforms which are both inclined at an angle. The rope has uniform mass density, and its coefficient of friction with the platforms is 1. What is the largest possible fraction of the rope that does not touch the platforms?

There are N nodes, connected by edges. There is exactly one path from any node to any other node. Then, something occurs and each node has a probability p of disappearing. What is the maximum for the expected value of the number of connected components that results, and for what value of p is this maximum achieved?

A point particle of mass m sits at rest on top of a frictionless hemisphere of mass M, which rests on a frictionless table. The particle is given a tiny kick and slides down the hemisphere. At what angle (measured from the top of the hemisphere) does the particle lose contact with the hemisphere?

Each of the integers from 1 to n is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, A, B, and C, take turns in the order A, B, C, A, ... choosing one card at random from the deck. After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered 1. Show that for each of the three players, there are arbitrarily large values of n for which that player has the highest probability among the three players of winning the game.

We take a trip down memory lane, analyzing the rational/irrational behavior of a function described as an infinite series.

Prove that a given sequence of rational functions are in fact all polyomials with integer coefficients.

A class with 2N students took a quiz, on which the possible scores were 0, 1, ..., 10. Each of these scores occurred at least once, and the average score was exactly 7.4. Show that the class can be divided into two groups of N students in such a way that the average score for each group was exactly 7.4.

Given N lines distributed randomly along the outside of a circle, we can ask some interesting questions.

Define a positive integer n to be squarish if either n is itself a perfect square or the distance from n to the nearest perfect square is a perfect square. For a positive integer N, let S(N) be the number of squarish integers between 1 and N, inclusive. Describe the behavior of S(N) as N approaches infinity.

A placeholder post for a future foray into the cunning world of algorithms for finding the closest two points amongst a finite set of points in a plane.

Suppose an integer N can be expressed as a sum of 2017 consecutive positive integers, but not as a sum of any other k consecutive positive integers. What is the smallest such N?

You are playing a coin flipping game. If you flip a head, you win. If you flip a tail, you now need to flip two heads in a row to win. In general, you keep track of the number of tails flipped, and you need to flip that many tails in a row plus one to win the game. What is the probability of winning the game?

If you are collecting cards from a set of size N, and you collect them in packs of c unique cards, what is the expected number of packs you will need to go through to collect them all?

JZ once asked us -- myself, Kevin, and Jeff, that is -- a now classic riddle. Given a balance and 3 uses of said balance, how can you always find the bad apple out of thirteen apples? The bad apple weighs slightly more, or slightly less, but you do not know which.