Apologies for not posting anything in the last few weeks, was traveling on work or vacation through this period.

This was not a very difficult puzzle, but I got correct answers only from the two people who solve every puzzle – Pratik Poddar and Suman Saraf – thank you and well done!

(a) Color the board like a chessboard, i.e. alternate black and whites. Notice that a bug on a white square can only travel to a black and vice versa. Since there are a total of 81 squares, there will be one more white square (or one more black square). Hence when the bugs move, there will correspondingly be at least one empty white square (or black square).

(b) Bugs can change places pairwise. One can leave only the bug in the centre square or the corner, which will be the only empty square.

Hope you all enjoyed the puzzle!

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!