The Prince Rupert Problem

Since 2008 I’ve spent part of my free time coaching a couple of really great ultimate frisbee teams:  Boston’s Brute Squad and Seattle’s Riot.    Other than one player on Brute Squad who was getting her PhD in math from Cornell, there hasn’t been a lot of overlap between my math activities and my ultimate frisbee activities.

BUT a few days ago Callie Mah from Riot sent me a link to a math book that she thought I’d be interested in:

Image

I bought it yesterday up in Boston, and Callie was  absolutely right – what a wonderful book for showing fun math to a general audience!  This morning I decided to open it up to a random page and do a fun little project with the boys on whatever topic was being discussed on that page.  We landed on the “Prince Rupert Problem.”

I do not remember hearing about this problem previously, but luckily it is pretty easy one to understand:  What is the largest cube that can pass through a cube of a given size?

The problem has a neat history.  It came out of some sort of bet that Prince Rupert made,   and took about 100 years to solve.   Surprisingly, there wasn’t a lot written about the problem on the internet, and the easiest reference for the history of the problem seems to be the Wikipedia page:

http://en.wikipedia.org/wiki/Prince_Rupert%27s_cube

The approach I took with my kids for the project this morning was to use our Zometool kit to look at a similar problem for a square first. – what is the largest square that can pass through a square of a given size?  The Zometool squares worked really well in helping us compare the relative size of the two squares:

Next we moved to the slightly more complicated problem of finding the largest square that could pass through a given cube.  This problem is a great introduction to visualizing 3D geometry, and once again the Zometool kit helped tremendously with the visualization:

Finally we moved on to the actual Prince Rupert Problem – what is the largest cube that can fit through a cube of a given size?  The solution – a cube with sides of length \frac{3 \sqrt{2}}{4} times the side length of the other cube (or about 1.06 times the side length) – was a little too difficult to cover in a short movie so I just mentioned the answer and showed how to think about it.  By coincidence we had two Rubik’s cubes with sizes in almost exactly the right proportion, so we had a nice visualization here, too

All in all, a really fun project.  Thanks Callie!!

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