Question 9.3.1

There are eight postage stamps in a bag: four red ones and four green ones. Three people each pick two of these stamps and stick them to their own forehead, without looking at the stamps, so everyone can see each other’s stamps, but not their own. The first person says ‘I don’t know what colour my stamps are.’ The second person says, ‘I don’t know either.’ And the third says, ‘I don’t know either.’ Then the first person says, ‘now I know.’

What colour were the stamps on the first person’s forehead? You can assume that all the people were telling the truth, and they were very good at logic, so that if they had enough information to work out the colour of their stamps, they would know. (This question comes from one of Martin Gardner’s books of mathematical puzzles, which I highly recommend.)

Assuming that the only possible answers are ‘two red stamps’, ‘two green stamps’, or ‘a red stamp and a green stamp’, can you work out which one it is without actually solving the logic puzzle? Show hint

Show answer

There are the same number of red stamps as green stamps, so if we swapped the colours, the question would still be the same. You could imagine swapping the words ‘red’ and ‘green’ so ‘four red ones and four green ones’ becomes ‘four green ones and four red ones’, which is the same thing because the order doesn’t matter. In other words, the question is symmetric with respect to swapping the colours. Therefore the answer must also be symmetric in this way, so it has to be ‘a red stamp and a green stamp’.

If the question had had three red stamps and five green stamps, then it wouldn’t be symmetric and ‘two red stamps’ or ‘two green stamps’ would be sensible answers, but because there was no difference between red and green in the question, there shouldn’t be a difference between red and green in the answer.