# combinatorics

## Van der Waerden Theorem

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

## Dinner Party Problem

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

## 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...

## Lucas’ Theorem

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

## 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...

## Envy-free Cake Division

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

## Farmers and Pesky Birds

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

## 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...