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This question already has an answer here:

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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marked as duplicate by Bass, Rubio Jul 12 '18 at 15:57

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    $\begingroup$ Do you have an answer? Because I don't think it's possible with the simultaneous guessing $\endgroup$ – Chris Cudmore Jul 12 '18 at 15:24
  • $\begingroup$ 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. $\endgroup$ – Bass Jul 12 '18 at 15:47
  • $\begingroup$ I have heard of this puzzle. I think a similar version was in the AMC (Australian Maths Competition). $\endgroup$ – Mr Pie Jul 13 '18 at 1:48
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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.

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    $\begingroup$ 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. $\endgroup$ – hexomino Jul 12 '18 at 15:54
  • $\begingroup$ @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. $\endgroup$ – hagfy Jul 12 '18 at 16:14
  • $\begingroup$ Ah I see, I made an error in my calculations. $\endgroup$ – Liora Haydont Jul 12 '18 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 $\endgroup$ – Saeïdryl Jul 12 '18 at 15:20
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    $\begingroup$ How does person X know to say 6? $\endgroup$ – Chris Cudmore Jul 12 '18 at 15:27

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