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Alternating T-Shirt Problem

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Ten friends walk into a room where each one of them receives a hat. On each hat is written a unique real number. Each person can see the numbers written on his friends' hats, but not her own. The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt. Wearing the respective T-shirts, the friends gather again and are lined up in the order of their hat numbers. The desired property is that the T-shirts colors now alternate. They can develop a strategy beforehand, but then no information is passed beyond the numbers observed.
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