WIM Video: The Pigeonhole Principle

The latest in Elisenda Grigsby’s WIM Video series features Harvard graduate student and Girls’ Angle mentor Charmaine Sia.  In it, she gives two applications of the pigeonhole principle:

If you’d like to try to solve the problems that Charmaine solves in this video before watching, here they are:

Problem 1. Given 5 points on a sphere, show that there is a hemisphere that contains 4 of them. (For this problem, if a point is on the boundary of the hemisphere, then that point is contained in the hemisphere.)

Problem 2. Given a sequence a_1, a_2, a_3, . . . , a_n of integers, show that there exists a sum of the form a_x + a_{x+1} + a_{x+2} + \dots + a_y which is divisible by n.

The pigeonhole principle is a very basic mathematical principle and appears so often in mathematical arguments that it is often used without even being pointed out. I remember my first formal encounter with it was in the following problem:

Given that more than 5 million people live in New York City and that nobody has more than a million head hairs, prove that there are at least two people in New York City with exactly the same number of hairs on their heads.

The WIM video series was conceived of by Girls’ Angle director Elisenda Grigsby. The series features women in mathematics presenting pieces of math that excited them when they were in middle and high school. The series began with funding from Science House and continues with funding from the Big George Ventures Philanthropy Fund.

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