Yet another very fine puzzle from *A Moscow Math Circle: Week by Week Problems* by Sergey Dorichenko.

Each square of a 9 x 9 board has a bug sitting on it. On a signal, each bug crawls onto one of the squares which shares a side with the one the bug was on. (a) Prove that one of the squares is now empty. (b) Can the bugs move so that there would be exactly one empty square?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. 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.

Happy bugging!

From square marked 1 to square marked 64, the maximum manhattan distance possible between the two squares is 14 and the maximum difference between any two numbers is 63. Since 63/14>4, there has to be at least one step where the jump is greater than 4. Hence proved.

Hope you all enjoyed the puzzle!