A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number....

Continue reading...# combinatorics

## Euler Characteristic

Take out a sheet of paper. Pop quiz! (just kidding). Draw any number of dots on your page. Now connect...

Continue reading...## Matching Problem

Suppose I return N homeworks randomly to my N students. What is the chance that no student gets back her...

Continue reading...## Toggling Light Switches

Imagine 100 light bulbs with light switches numbered 1 through 100, all in a row, all off. Suppose you do...

Continue reading...## Leapfrog Addition

Here’s a nice mathematical magic trick based on properties of the Fibonacci sequence. Give your friend a card with ten blank lines, numbered 1...

Continue reading...## Proofs without Words

There are many facts which can be proved just by looking at a picture! For instance:x2 – y2 = (x+y) (x-y)is illustrated...

Continue reading...## Pascal’s Triangle

Consider the triangle in Figure 1, called Pascal’s triangle. It consists of numbers where each entry is the sum of the...

Continue reading...## Pigeonhole Principle

Here’s a challenging problem with a surprisingly easy answer: can you show that for any 5 points placed on a sphere,...

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