The 3rd video in Kelsey Houston-Edwards’s amazing new series was published last week. I’ve already used the first two videos for projects with the boys – I love this series so much!
the latest video is about the Pigeonhole Principle and begins with the question – Do any two human beings have exactly the same number of body hairs:
Before diving into the video I asked the boys what they thought about the hair question – fortunately I got two different answers!
Next we watched Houston-Edwards’s new video:
Here’s how the boys reacted to the video:
(1) They were excited about the hair result and were also able to understand and explain it.
(2) They gave a nice summary of the Pigeonhole Principle.
(3) They really liked the example about 5 points on a sphere, so we took a really close look at that example. One of the tricky parts of that problem is understanding *why* you can draw an equator through any two points – both kids gave nice explanations of that idea.
Now I moved on to a couple of fun Pigeonhole Principle examples that weren’t covered in the video. I wanted to show the boys that the idea comes up in lots of different situations, including some that are not at all obvious Pigeonhole Principle situations!
The first example comes from my college combinatorics textbook – Applied Combinatorics with Problem Solving by Jackson and Thoro:
Small twitter math world fun fact – the professor for this class (~25 years ago!) was Jim Propp!
Here’s the problem (which is example 5 on page 35 of the book):
Suppose that we are given a set X of 10 positive integers, now of which is greater than 100. Show that there are two disjoint nonempty subsets of this set whose elements have the same sum.
I had to do a little bit of work on the fly to translate the problem into something that the boys could understand (and also explain quickly why there are 1024 subsets), but it seemed like they enjoyed this example:
The last problem is one I remembered when reading through some of the other examples in Jackson and Thoro’s book and is one that I talked about with the boys last year:
Here’s the problem:
Show that every positive integer has a multiple whose base 10 representation consists of only 1’s and 0’s.
It certainly isn’t obvious at all at the start why this is a Pigeonhole Principle problem!
As I said at the beginning – I love this new series from Kelsey Houston-Edwards. I’m so happy to be able to use these videos to explore fun mathematical ideas with my kids!