# The Mathematics of Various Entertaining Subjects volume 3

The third of the biennial MOVES (Mathematics of Various Entertaining Subjects) conferences was organized in 2017, as usual in the MoMath museum in New York. This book contains its proceedings. Also volume 2 was reviewed here earlier. The editors, the publisher, and the concept of the previous volumes did not change. The overall theme of the conference was this time *The magic of math* but that can of course include about anything. There are four parts, each containing three to six papers: (1) Puzzles and brainteasers, (2) Games, (3) Algebra and number theory, (4) Geometry and topology. I will pick some examples to give an idea about the contents.

The six papers in the part *Puzzles and brainteasers* usually start by challenging the reader with a number of problems for which the solution is given at the end. An example: a variant of a well know prisoner's hat riddle can be formulated as follows: in a set of prisoners each prisoner gets either a red or a yellow hat. They see the colour of the hats of all the others but they do not know the colour of their own hat. They have to answer simultaneously either the colour of their own hat or pass. If one guesses the wrong colour or if all pass, their sentence is extended. If one guesses correctly and none is wrong, they are freed. What should be their strategy (supposing they want to be free)?. To solve this problem the colours are represented as 0 or 1. Each prisoner can compose the 0-1 string of the ordered set of prisoners, except for his own bit. This can be treated like an error correcting code problem. There exists a set such that the probability that the binary string belongs to that set is much larger than that it belongs to the complement of that set. So the problem is reduced to computing a Hamming distance. The solution is explained in elementary steps, but it eventually ends with explaining Hamming codes, finite fields, parity check matrix, Hamming distance, etc. Another guessing problem involves random walks and hidden variables. So, this shows that what starts as a puzzle or game, will eventually lead to the introduction of some mathematics, which is the set up of almost all the contributions in this book.

In the *Games* section we find five papers. Intriguing questions, sometimes with surprising answers, can be asked. Take for example the following ones. What are the chances to have a winning row, column of diagonal in a Bingo game (American style) played with 5 x 5 cards with or without the central square free? How to code and count different Tsuro cards? How difficult is it to loose a checkers game? (Suppose you want your son to win without violating the rules.) Here is another problem involving probability: Each player has to move in turn a random number of steps along a path of squares and the target is to end exactly at the end square. If they don't, they have to turn back on the path until they have made the required number of steps. How many moves are needed on average to finish? How many times is a square visited? Several versions exist of another game called "Japanese ladders" (also known as "Ghost Leg", or with several other names). It is a challenging problem to find a strategy to play these games by adding rungs or legs and win. The mathematical equivalent is to decompose and manipulate permutations as a succession of adjacent transpositions. Answering all these questions in the Games section, involves combinatorics, probability, symmetry, graphs, etc.

The mathematics required in the part on *Algebra and number theory* is obvious from its title. The first long paper is by Persi Diaconis and Ron Graham. Diaconis is a well known mathematician and magician and he was an invited speaker at the 2017 MOVES conference. The contribution is about the magic of Charles Sanders Peirce, known to be the father of pragmatism. Peirce's 1908 paper *Some amazing mazes* is difficult to read, so here Diaconis and Graham analyse the principles that support one of the most complicated card tricks ever, each of these principles can be inspiration for some card trick in its own right. The mathematics involved is interesting as well. The analysis contains for example an implicit proof of Fermat's little theorem. Other papers in this section also involve symmetry, groups, modulo arithmetic, and graphs, to solve games like Khalou, or puzzles like KenKen.

The three papers in the last part are grouped under the title *Geometry and topology*. One paper is about flexagons, knots, and twisted bands (of which the Moebius band is the simplest example). The second is an interesting graph problem. Consider a regular triangular grid graph covering the plane with cities located on some of the vertices. What is the shortest set of railroad tracks that allows to reach every city from any other city if the railroad can only move along grid lines? This problem is originating from a board game called TransAmerica, but it is also a practical problem to wire the lights on the grid points of the glass dome of the Dalí museum in St Petersburg, Florida which is constructed as a triangular metal grid frame filled with 1100 triangular windows of approximately equal size (although no two are identical). The last paper in this section is about the incredible amount of ways in which a set of Lego blocks can be connected. Just 8 simple jumper plates (one notch at the top and three slots at the bottom) allow for 393314 different compilations. What is the entropy, i.e., hat are the asymptotics of $\frac{1}{n}\log N(n)$ as $n$ becomes large and $N(n)$ is the number of ways to compile $n$ elements?

As this incomplete survey illustrates, this is a mixture of fun and serious mathematics where professional mathematicians, computer scientists, and enthusiastic gamers and puzzlers can meet. Recreational mathematics has grown out of its infancy and there are some tough results that can be proved and some serious challenges that can be formulated as open, yet unsolved, problems. Some games and puzzles may have been invented by smart amateurs for recreation, solving the puzzle, winning the game, or computing your chances, has become a topic that often requires some mathematical training. Anyone from amateur to professional will be fascinated by the diversity of challenges and solutions proposed. Not the highest level of mathematical abstraction is needed, so with some elementary knowledge the book can we assimilated, but still, it requires a certain willingness to wade through all the mathematics, which is intended to be an essential part. But mathematics is fun and the book is playful and accessible. Moreover, as Bhargava writes in his foreword: isn't most, if not all, mathematics recreational?

**Submitted by Adhemar Bultheel |

**9 / Sep / 2019