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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