Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In each square in a table $10\times 10$ we write a whole number so the difference between any two numbers by the neighboring squares must be no more than $5$ (two squares are neighboring if they have a common side). Show that two of the numbers must be equal. Please help me!

share|cite|improve this question

Suppose all the numbers were different (say, numbered from 1 to 100, because anything farther is automatically impossible if that case is). How would you place the numbers down to minimize their differences? Are all these differences less than 5? Can you prove that it's impossible?

Solution: Write the number 1 on the top-left corner, and the number 100 on the bottom-right. The distance between 1 and 100 is 18 squares. So, trace out any path from 1 to 100 that is 18 squares long. Since the difference between consecutive squares is at most 5, the difference between the first and last squares along this path is at most 90, so a difference of 99 (from 1 to 100) is impossible.

share|cite|improve this answer
Please give me a complete demonstration. – Manuel Feb 28 '13 at 21:08
Done. ${}{}{}{}{}{}{}$ – Joe Z. Feb 28 '13 at 21:21
Well done, Joe --- now Manuel can just hand this in as his own work, without putting even a smidgen of effort into it, or having learned anything at all from it. He'll be back for more. – Gerry Myerson Feb 28 '13 at 21:44
I was simply assuming good faith. He had two hours to do it himself. – Joe Z. Mar 1 '13 at 0:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.