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!
Mike's Math Page 
I really liked this problem, and I’m making a note to share it with my geometry students. Thanks!