"100 gnomes are lined up in order of decreasing height, and then each randomly given either a black or white hat. Each can see all the hats of the gnomes in front of him, but not any behind, nor his own. Starting with the rearmost, an executioner asks each gnome what colour his hat is. If the answer is incorrect, the gnome is killed. The other gnomes can hear the answer, but not whether it was correct. Assuming the gnomes are allowed to confer in advance, how many can survive?" Can be extended to N colours of hats without too much difficulty.