600 cell

[David Ash]

One more problem proposal:

Give, with proof, the distance between two opposite vertices of a 600-cell (regular polytope) whose edge length is 1.

[t...@bellefleurbooks.com]

Dear David:

Wikipedia can be a lot of help proving this one.

https://en.wikipedia.org/wiki/600-cell

Sincerely,

Tom

On 2021-09-28 20:21, 'David Ash' via Competition Corner Participant

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[David Ash]

If I were grading this, and having just looked at the Wikipedia article, my thought is that the Wikipedia article provides (most of) the answer but (almost none of) the proof. Additionally, if I were grading this, I'd grade a proof that derives the coordinates from first principles, rather than just verifying coordinates from a Wikipedia article, higher. I want a proof that shows someone has thought relatively deeply about the structure of the 600-cell, rather than performing the last couple of steps of a computation from a Wikipedia that provides (without rigorous proof) the coordinates.

But this does raise a meta-question: for a month-long competition, how do we ensure that someone doesn't just find the solution to the problem online somewhere? Even if we think the problem is new, there might be a web page out there somewhere with the problem or something very similar. How do we avoid this becoming simply a test of who has better web search skills? How does USAMTS address this issue?

[David Ash]

Or better yet a completely geometrical proof that doesn't rely on reducing everything to coordinates. For a journal format, it seems to me we should be looking for more than just the right answer--esp for the higher grades of high school. For the lower grades it will probably be sufficient just to come up with the right answer.