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Riddle: The Hat Guessing Game

Three kids, Writey, Fruity, and Spicy, are standing around. Their friend, Bob, proposes a challenge. The three kids will close their eyes, and Bob will go around and put one of two hats on each of them, as follows:

  • On Writey, Bob will put a hat with a picture of either a pencil or a pen
  • On Fruity, Bob will put a hat with a picture of either an apple or an orange
  • On Spicy, Bob will put a hat with a picture of either a salt-shaker or a pepper-shaker

Bob will pick the hat to put on each kid completely randomly (50% chance of either hat). Then, the kids will open their eyes. At this point, each kid will be able to see the hats of the other two kids, but won't be able to see his own hat. The kids cannot give each other ANY indication about what is on the other kids' hats.

Now, on the count of three, simultaneously, each kid will either say a guess for the picture on their hat, or say 'pass'. Specifically, on the count of three:

  • Writey will either say 'pencil', 'pen', or 'pass'
  • Fruity will either say 'apple', 'orange', or 'pass'
  • Spicy will either say 'salt', 'pepper', or 'pass'

Note that the kids say their guesses (or 'pass') at the exact same time and thus cannot give any information to the other kids. To clarify, the only information that each kid has when saying his guess is the pictures on the other kids' hats.

The kids all win the game if at least one person guesses the item on their own hat correctly, as long as nobody guesses the item on their hat incorrectly. For example, if two of the kids say 'pass' and the third kid guesses his hat item correctly, they all win the game, but if two of the kids guess their hats correctly but the third guesses incorrectly, they lose. If all three kids pass, they also lose.

Before the game starts, the kids are allowed to come up with a strategy in order to try to maximize their chances of winning. Bob notes that it's impossible to guarantee a win every time, and that there are simple strategies to ensure the kids will win 50% of the time (Writey and Fruity could always say 'pass', and Spicy could always say 'salt'...then, since Spicy's hat will have a salt-shaker on it 50% of the time, they'll win 50% of the time). But Bob tells them they can do much better than 50%.

What strategy should they use to substantially increase their chance of winning?

The strategy should be such that each kid always knows exactly what they will say based on the two hats they can see.

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