I am so used to wise-men puzzles about hats, that I was pleasantly surprised when Leonid Makar-Limanov gave me a wise-men puzzle that didn’t include them.
A sultan decides to check how wise his two wise men are. The sultan chooses a cell on a chessboard and shows it to the first wise man. In addition, each cell on the chessboard either contains a rock or is empty. The first wise man has to decide whether to remove one rock or to add one rock to an empty cell. Next, the second wise man must look at the board and guess which cell was chosen by the sultan. The two wise men are permitted to agree on the strategy beforehand. What strategy can they find to ensure that the second wise man will always guess the chosen cell?
If you are stuck, there are many approaches to try. You can attempt to solve the puzzle for a board of size 1 by 2, or for a board of size 1 by 3. Some might find it easier to solve a version in which you are allowed to have a multiple number of rocks on a cell, and the first wise man is permitted to add a rock to a cell that already contains rocks.