10 people will stand in a circle, and each have a hat placed on their head. The hats will each be labeled with a digit between 0 and 9, with repeats allowed. They will be able to see everyone's hat but their own. After looking around, they must all simultaneously guess the digit on their hat. Before this happens, the team may agree on a strategy, but once the game begins, they may not communicate in any way. They win as long as at least one person guesses correctly. Show how the team can guarantee victory.
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1$\begingroup$ Do you have an answer? Because I don't think it's possible with the simultaneous guessing $\endgroup$– Chris CudmoreCommented Jul 12, 2018 at 15:24
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$\begingroup$ This is a duplicate of the previous question in the hat-guessing tag, which itself is was closed as duplicate of the more general version. $\endgroup$– BassCommented Jul 12, 2018 at 15:47
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$\begingroup$ I have heard of this puzzle. I think a similar version was in the AMC (Australian Maths Competition). $\endgroup$– Mr PieCommented Jul 13, 2018 at 1:48
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2 Answers
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Strategy
Let the sum of the numbers on all ten hats be $S$.
To each person, assign a unique digit between $0$ and $9$. When it comes to guessing their own hat number, each person picks the number which allows the last digit of $S$ to correspond to their assigned digit. Exactly one of them will be right.
answered Jul 12, 2018 at 15:33
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1$\begingroup$ Okay, for those, say we assign 0 to 9, left to right. Then the idea for each person is to get the total sum's last digit to match their assigned digit so they will say (left to right) 5, 8, 2, 8, 1, 2, 8, 6, 9, 5 and only the last person is right. This happens because the last digit of the total sum is 9. $\endgroup$– hexominoCommented Jul 12, 2018 at 15:54
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$\begingroup$ @LioraHaydont They take the sum of all digits they can see, they don't know the total beforehand. In your example, the total is 49, but the 10th guy can only see 44 (4+6+9+4+6+6+1+8+0). Since his assigned number is 9, he guesses 5 (as 4 + 5 = 9), which happens to be his hat number. $\endgroup$– hagfyCommented Jul 12, 2018 at 16:14
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$\begingroup$ Ah I see, I made an error in my calculations. $\endgroup$ Commented Jul 12, 2018 at 16:22
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If
everybody says the number on person "X"'s hat (predetermined who "X" is before, during strategy), then person "X" will be right, and everyone wins.
Thanks to Chris Cudmore & Saeïdryl I see that this won't work as I originally thought.
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$\begingroup$ I think "they must all simultaneously guess the digit on their hat" implies they all speak at the same time, but I'm probably wrong $\endgroup$– SaeïdrylCommented Jul 12, 2018 at 15:20
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2$\begingroup$ How does person X know to say 6? $\endgroup$ Commented Jul 12, 2018 at 15:27