Weihnachtsmann rot

Challenge from 12. December

Hat Challenge 2019

Authors: Aart Blokhuis, Gerhard Woeginger

Project: 4TU.AMI

Challenge:

Santa Claus says to Atto, Bilbo, and Chico, “My dear super-smart elves! As every year, the Mathematical Advent Calendar is going to pose a tricky puzzle with colored hats on the heads of elves. For this reason, I will invite you for coffee and cake tomorrow afternoon.”
“Great, we love to come!”, answer Atto, Bilbo and Chico.

Santa Claus is pleased, “Well, then tonight I am going to prepare three blue hats, three yellow hats, and three red hats. Tomorrow, I will put one hat on each of your heads such that none of you can see his own hat color. However, you will see the hat colors of the other two elves, but you are not allowed to exchange any information with each other.”

Santa continues: “Then, simultaneously, each of you must show a certain number of fingers.

  • A single finger means that you guess that your own hat is blue,

  • two fingers mean that you guess yellow,

  • and three fingers mean that you guess red.

  • Zero or four or more fingers mean that you prefer not to guess.

If at least one of you guesses wrong, the game is over and you have to leave. If none of you guesses wrong and at least one of you guesses the right color, then you get a tasty piece of Sacher torte and a nice cup of coffee.”

Atto wants to know, “How are you going to choose our hat colors?”
“They are chosen randomly such that each of the 27 color combinations arises with the same probability”, answers Santa Claus.

Bilbo asks, “Will we be able to see the six hats that you do not use?”
“No!”, answers Santa Claus, “The unused hats will be hidden in the closet!”

And Chico asks, “What is going to happen, if none of us guesses?”
“Then, the game is over and you have to leave”, says Santa Claus, “Coffee and cake will be served only if at least one of you guesses right and none of you guesses wrong. Do you understand?”

The elves start to ponder. They discuss and they think. They think and they discuss. Then, they discuss some more, and then they think some more. Eventually they manage to develop an amazing strategy that maximizes the number M of color combinations for which coffee and cake will be served.

The question is of course: How big is this number M?

Artwork: Friederike Hofmann

PDF download

Possible answers:

  1. M = 9.

  2. M = 10.

  3. M = 11.

  4. M = 12.

  5. M = 13.

  6. M = 14.

  7. M = 15.

  8. M = 16.

  9. M = 17.

  10. M = 18.

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