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This puzzle appeared in the 'Skeptical Adversaria' (the newsletter of ASKE, the Association for Skeptical Enquiry), January 2006.
You are standing in a field with, let’s say, 100 other people. Connecting this field with the adjacent one is a narrow style. Everyone is to proceed over the style into the adjacent field. As each person is climbing over the style either a red or a green hat is placed on his or her head. As they enter the second field, people must immediately divide off into those with red hats and those with green. No one can see his or her own hat (directly or through a reflecting surface) and no one, either in the group or observing, must inform other people of the colour of their hat.
What simple instruction could you give to the group on how they can perform this task?
Once the whole task is completed, is it guaranteed that everyone will know the colour of his or her hat?
The solution to the puzzle is as follows.
The instruction is as follows. As people stream into the second field they form a row. From the third person onwards the rule is that if the hats of the people already in the row are all red or all green the next person goes to one end of the row (it doesn’t matter which). If some are red and some are green he or she stands between the two people with different coloured hats. Following this rule from the start will mean that all the people with red hats will be on one side of the line and all the people with green hats will be on the other.
The last person in the field will not know the colour of his hat. If all 99 of his colleagues have the same coloured hat he goes to one of the ends of the row but still cannot tell which is the colour of his own hat. (If the person before him went to the other end of the row, that person will realise he or she has the same hat as the adjacent person when the last person takes his place.) If some of the 99 have red hats and some green hats then the last person goes between the two adjacent people whose hats differ. At that point everyone, including his two neighbours, will know the colour of his or her hat but for the last person, his hat could be either colour from the information he has.
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