My son was working on some old AMC 10 problems today and this problem gave him a bit of trouble. The problem is #18 on the 2007 AMC 10 b:

We’ve studied a similar problem before in these three projects:

A Strange Problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s n-dimensional sphere problem with kids

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

Related problems seem to be following me around this week, though. I saw one version when I bought Martin Gardner’s book at the Omaha public library book sale:

And another version when I watched Grant Sanderson’s latest video:

For tonight’s project we worked through the problem and talked about some of the fun ideas that come into the problem in higher dimensions.

Here’s the introduction to the problem:

Next my older son showed his solution to the problem. There were a few stumbles, but he got through it:

Next I had my younger son try to solve a similar problem – he also struggled a bit but eventually found the ideas necessary to solve it:

The next problem we looked at was the version of the problem in 3 dimensions. Once we solved that problem the boys had a good guess of how the problem would look in 4 dimensions!

Finally we played around with the strange ideas that come into play in 10 and 1206 (though I think we said 1063 in the video):

So, definitely a fun problem – from a nice 2d geometry problem to totally non-intuitive problem about higher dimensional spheres. Funny to run into it three times in the last week!