In Jan 2023 I went to the Joint Math Meeting of the AMS and the MAA and took notes on things to look up later. In one of the talks they discussed a problem and indicated that the answer was known, but did not give a reference or a proof. I emailed the authors and got no response. I tried to search the web but could not find it. SO I use this blog post to see if someone either knows the reference or can solve it outright, and either leave the answer in the comments, point to a paper that has the answer in the comments, or email me personally.

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A chessboard has squares that are 1 by 1.

Pennies have diameter 1.

QUESTION:

For which n is there a way to place n pennies on squares of the n x n chessboard so that all of the distances between centers of the pennies are DIFFERENT?

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I have figured out that you CAN do this for n=3,4,5. I THINK the talk said it cannot be done for n=6. If you know or find a proof or disproof then please tell me. I am looking for human-readable proofs, not computer proofs. Similar for higher n.

I have a writeup of the n=3,4,5 cases here (ADDED LATER- I will edit this later in light of the very interesting comments made on this blog entry.)

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With technology and search engines it SHOULD be easier to find out answers to questions then it was in a prior era. And I think it is. But there are times when you are still better off asking someone, or in my case blog about it, to find the answer. Here is hoping it works!

ADDED LATER: Within 30 minutes of posting this one of my readers wrote a program and found tha tyou CAN do it for n=6 and gives the answer. Another commenter pointed to a website with the related quetion of putting as many pawns as you can on an 8x8 board.

ADDED LATER: There are now comments on the blog pointing to the FULL SOLUTION to the problem, which one can find here. In summary:

for n=3,...,7 there IS a way to put n pennies on a chessboard such that all distances are distinct.

for n=8,...,14 a computer search shows that there is no such way.

for n=15 there is an INTERESTING PROOF that there is no such way (good thing - the computer program had not halted yet. I do not know if it every did.)

for n\ge 16 there is a NICE proof that there IS such way.

I am ECSTATIC!- I wanted to know the answer and now I do and its easy to understand!