Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in...

Continue reading...# combinatorics

## Dinner Party Problem

How many people must you have at dinner to ensure that there are a subset of 3 people who all...

Continue reading...## Continuum Hypothesis

We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with...

Continue reading...## Lucas’ Theorem

Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p...

Continue reading...## Cantor Diagonalization

We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as...

Continue reading...## Envy-free Cake Division

Say you and a friend wish to share a cake. What is a “fair” way to split it? Probably you...

Continue reading...## Farmers and Pesky Birds

Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in...

Continue reading...## Odd Numbers in Pascal’s Triangle

Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N...

Continue reading...## How many Rationals?

How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are...

Continue reading...## Dominoes on a Chessboard

A standard 8×8 chessboard can easily be covered (tiled) with non-overlapping dominoes (1×2 pieces): simply use 4 dominoes in each...

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