The Raven's Hat Fallen Pictures, Rising Sequences, and Other Mathematical Games

The Raven’s Hat explores games that at first glance appear to be impossible to solve, but actually have strategies for winning based in mathematics and statistics. The classic example of this, covered in the first chapter and illustrated in the title, is the hat color game. Often phrased in terms of prisoners and a warden, a finite group of people have a hat colored one of two possible colors placed on their heads by the warden. The players or prisoners can see the hats of the other people, but cannot see their own. They are not allowed to communicate with each other in any way after the hats are on their heads. Each prisoner must answer a question about the color of their hat. The answer can be either of the colors or ‘I don’t know.’ If at least one player guesses correctly and no other player guesses incorrectly, the prisoners escape and win the game. Otherwise if even one player guesses the wrong color or everyone refuses to guess a color, they all lose. You can see how this feels impossible: if they are not allowed to communicate, how can there possibly be a strategy that allows them to win most of the time? I’ll let you read the chapter to discover the optimal approach rooted in Hamming codes, but you’ll find that a group of 3 players playing with the optimal strategy will lose with probability \( 1/4 \) while a group of \( n=2^{m}-1 \) players will lose with the probability of \( 1/(n+1) \).

 

Another interesting chapter explains the mathematics of a card trick. This is the familiar magic trick where an audience member chooses a card and puts it back in a deck that is then shuffled and cut. It seems like there is no way for the magician to identify the card, but somehow the magician identifies the right card a surprising percentage of the time. The authors show how the magician can use rising sequences in permutations to determine the correct card. It is not a perfect strategy, with a probability of success about 84%, but the magician will have other performative tricks to employ if the statistical strategy fails. 

 

The structure of the book is consistent. Each chapter is self-contained and starts by setting up the game and giving the reader a chance to think about how they might play.  Basic strategies are usually provided to give upper and lower bounds on the probability of winning. Then the necessary math is introduced and connected to the game. Variations and generalizations such as random strategies versus adversarial opponents are often introduced.  The chapters conclude with some history of the problem with source material, as well as some practical options for actually playing the game.

 

If you are interested in a super-rigorous read, this is not that. Expect clear explanations with some of the technical mathematics left to the sources referenced in the book. The authors do provide three appendices for mathematical notation, definitions, and extra details not explicitly covered in the chapters. The book contains a decent reference list of the most relevant papers and books, as well as a topical index.

 

The Raven’s Hat is appropriate for mathematicians and statisticians of varied experience and for lovers of games. Those interested in the mathematics can learn about how more abstract topic like cyclic groups and projective geometry can be applied to a concrete games.  Further, while I can imagine playing one these games in a math club meeting, I could also play with my extended family who love more complex board games. Of course, I’ve read the book, so maybe that would be more than a little unfair!

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Tricia Muldoon Brown (tmbrown@georgiasouthern.edu) is a Professor of Mathematics at Georgia Southern University with interests in combinatorics, recreational mathematics, and sports.