This is another beauty of a puzzle from A Moscow Math Circle: Week by Week Problems by Sergey Dorichenko.
The numbers from 1 to 64 have been placed in each square on a chessboard such that each number appears exactly once. Prove that there will be two squares sharing an edge whose numbers will differ by more than 4.
As always, please send your answers as comments within the blog (preferred), or send an e-mail to firstname.lastname@example.org. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.