April 26th, 2019 : Identical Rainbows

[xoxoxoBruce]

The Skittles label reads...
“No two rainbows are the same. Neither are two packs of Skittles. Enjoy an odd mix.”

A mathematician/computer scientist/engineer/teacher at the “Possibly Wrong” website took umbrage with the slogan.
So taking five colors, and 2.17 oz per pack, he made calculations you can see at the link, and have monster explain them to you.
He figured there would be two packages alike in less than 500 packages.
But there is only one way to prove his calculations.

I purchased all of the 2.17-ounce packs of Skittles for this experiment from Amazon in boxes of 36 packs each. From 12 January through 4 April, I worked my way through 13 boxes, for a total of 468 packs, at the approximate rate of six packs per day.

1 Take a pack from the box, open it, and empty and sort the contents onto a blank sheet of paper.
2 Take a photo of the contents of the pack.
3 Record, with pen and paper, the number of Skittles of each color in the pack (more on this later).
4 Empty the Skittles into a bowl.
5 Repeat steps 1-4; after six packs, save and review the photos, recording the color counts to file, verifying against the paper record from step 3, and checking for duplication of a previously recorded pack.

So, what’s the point? Why bother with nearly three months of effort to collect this data? One easy answer is that I simply found it interesting. But I think a better answer is that this seemed like a great opportunity to demonstrate the predictive power of mathematics. A few months ago, we did some calculations on a cocktail napkin, so to speak, predicting that we should be able to find a pair of identical packs of Skittles with a reasonably– and perhaps surprisingly– small amount of effort. Actually seeing that effort through to the finish line can be a vivid demonstration for students of this predictive power of what might otherwise be viewed as “merely abstract” and not concretely useful mathematics.

Original prediction

Proof