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I'm having trouble with this question, which appears in Professor Higgin's problem collection. There is an answer, though I do not really understand it. The question goes somewhat like this :

In a 10 by 10 grid, assign each square in the 10 by 10 grid a number from 1 to 100 so that no numbers are repeated. It would be ideal to choose this arrangement so that the difference of adjacent squares is as small as possible. Find a numbering that has a maximum difference between adjacent squares of 10 (which I have done). Is it possible to do better?

I am having trouble with the part of the question that ask whether you can do better, which is to say, can the maximum difference between adjacent squares on the grid be 9 or lower?

Thanks for anyone who answers!

JC12
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  • More generally, if you fill an $n$ by $n$ grid with $n^2$ different integers, then there is always a pair of adjacent numbers with difference at least $n$. – WhatsUp Dec 22 '19 at 01:43
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    @WhatsUp Do you know of a better solution to the general case than my solution on the linked problem? – Calvin Lin Dec 22 '19 at 04:46
  • @CalvinLin No, I just read your solution there and see that it works under the more general assumption. That's a great answer (: Before seeing your answer, I myself thought about the problem for several minutes and didn't get anything. – WhatsUp Dec 22 '19 at 04:49
  • I’m still a bit confused about the solution to the problem. Also, is there a way to prove the general rule? – JC12 Dec 22 '19 at 08:18
  • @JC12 - Calvin Lin proved the general rule in the other thread on Dec 6. – Paul Sinclair Dec 22 '19 at 17:00
  • Thanks, finally understand the solution. Calvin explained it very well. – JC12 Dec 22 '19 at 20:36

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