Mathematical games and recreations

Mathematical puzzles vary from the simple to deep problems which are still unsolved. The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics. Number games, geometrical puzzles, network problems and combinatorial problems are among the best known types of puzzles.

The Rhind papyrus shows that early Egyptian mathematics was largely based on puzzle type problems. For example the papyrus, written in around 1850 BC, contains a rather familiar type of puzzle:

Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hectares of wheat. What is the total of all of these?

Fibonacci is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13,... where each number is the sum of the previous two. In fact a wealth of mathematics has arisen from this sequence and today there are lots of problems related to the sequence. Here is the famous Rabbit Problem.

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive?

Cardan (1501 – 1576) invented a game consisting of a number of rings on a bar.

Fig. 3.1

It appears in the 1550 edition of his book De Subtililate. The rings were arranged so that only the ring A at one end could be taken on and off without problems. To take any other off the ring next to it towards A had to be on the bar and all others towards A had to be off the bar. To take all the rings off requires (2 ⁿ ⁺¹ - 1)/3 moves if n is odd and (2 ⁿ ⁺¹ - 2)/3 moves if n is even.

The Thirty Six Officers Problem, posed by Euler in 1779, asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 6 square so that no rank or regiment will be repeated in any row or column. The problem is insoluble but it has led to important work in combinatorics.

Another famous problem was Kirkman's School Girl Problem. The problem, posed in 1850, asks how 15 school girls can walk in 5 rows of 3 each for 7 days so that no girl walks with any other girl in the same triplet more than once. In fact, provided n is divisible by 3, we can ask the more general question about n school girls walking for (n - 1)/2 days so that no girl walks with any other girl in the same triplet more than once. Solutions for n = 9, 15, 27 were given in 1850 and much work was done on the problem thereafter. It is important in the modern theory of combinatorics. Around this time Sam Loyd's most famous game was the 15 puzzle.
Fig.3.2
It illustrates important properties of permutations.

The most famous of recent puzzles is the of Rubik's cube invented by the Hungarian Ernö Rubik. It's fame is incredible. Invented in 1974, patented in 1975 it was put on the market in Hungary in 1977. However it did not really begin as a craze until 1981. By 1982 10 million cubes had been sold in Hungary, more than the population of the country. It is estimated that 100 million were sold world-wide. It is really a group theory puzzle, although not many people realize this.

The cube consists of 3 3 3 smaller cubes which, in the initial configuration, are coloured so that the 6 faces of the large cube are coloured in 6 distinct colours. The 9 cubes forming one face can be rotated through 45 . There are 43,252,003,274,489,856,000 different arrangements of the small cubes, only one of these arrangements being the initial position. Solving the cube shows the importance of conjugates and commutators in a group.