New York Math Circle Fairfield County MTC

Knights and Knaves, Hats and Prisoner Puzzles

Knights and Knaves

Raymond Smullyan is famous for his Knights and Knaves problems. Knightsalways tell the truth, Knaves always lie, and you can’t tell just by looking atthem. Your task is to distinguish Knights from Knaves.

1. Three people, A, B and C, are just standing around, when you ratherbluntly ask A, “Are you a knight or a knave?” You can’t quite hear A’sresponse, so you ask B, “What did A say?” B replies, “A said he is aknave.” Suddenly, C blurts out, “Don’t believe B, he is lying!”

What are A, B and C?

2. Two people, A and B, are lounging around, when A states that, “Atleast one of us is a knave.” What are A and B?

3. Another two people, C and D, are present. C says, “Either I am aknave or D is a knight.” What are C and D?

Hats

Red and blue hats are placed on blindfolded players’ heads. The blindfoldsare simultaneously removed and the players can see the others’ hats, but nottheir own. Often, they have time to agree on a strategy in advance, which isgood, since they are not allowed to communicate during the game.

1. There are two players, each wearing a red or blue hat. They simulta-neously guess a color, and win if at least one of them guesses their ownhat color correctly. Can they strategize a win?

2. Three players are each wearing either a red or a blue hat. At the signal,each raises her hand if she sees at least one red hat. Armed with thisinformation, can at least one player be sure of the color of her hat?

3. Three players each wears a red or a blue hat. After the blindfolds areremoved and they see the other two hats, each is discreetly asked toguess the color of her or his hat, or to pass. If at least one playerguesses correctly and none of the guesses are incorrect, they win.

What strategy could the three players agree on in advance to maximizetheir chance of winning?

Japheth Wood <japheth@nymathcircle.org> July 16, 2012

New York Math Circle Fairfield County MTC

Prisoners

Prisoner puzzles are organized by a malicious warden, and the consequencesfor losing them are quite dire. Otherwise, they are similar to hat puzzles!

1. The warden lines up six prisoners, all facing forward, and places redand blue hats on their heads. When the blindfolds are removed, theprisoner in the back of the line can see the five hats in front of him.The next prisoner can see the four hats in front of him, and so on. Theprisoner at the beginning of the line can’t see any hats at all.

Starting at the back of the line, each prisoner is asked in turn to guessthe color of his hat. An incorrect guess warrants a severe whipping,otherwise just a slap on the wrist, which all the other prisoners hear.What strategy can the prisoners agree on in advance to minimize thepunishments received?

2. The warden selects 100 prisoners and places their names in 100 boxes(one name per box) that are lined up in the yard. Prisoners are broughtout one at a time, and allowed to look in 50 boxes for their name. Ifeach prisoner finds his own name, then all are freed. If even one prisonerdoes not find his name, they all suffer greatly. What strategy can theprisoners agree on in advance to minimize their suffering?

3. Infinitely many prisoners (named #1, #2, #3, . . . ) have red and bluehats placed on their heads. The blindfolds are removed, they lookaround, and each guesses the color of his own hat. Is there a strategythat guarantees that at most finitely many prisoners guess incorrectly?

References

[1] R. Smullyan, What is the name of this book, Prentice Hall, NJ, 1978.

[2] E. Brown and J. Tanton A Dozen Hat Problems, Math Horizons, April2009, 22–25.

[3] M. Gardner, Penrose Tiles to Trapdoor Ciphers, W.H. Freeman, NY,1989.

[4] P. Winkler, Mathematical Mind-Benders, A.K. Peters, Wellesley, MA,2007.

Japheth Wood <japheth@nymathcircle.org> July 16, 2012