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.
You can see a picture of the Rhind papyrus at THIS LINK.
Greek mathematics produced many classic puzzles. Perhaps the most famous are from Archimedes in his book The Sandreckoner where he gives the Cattle Problem.
You can see further details at THIS LINK.
Archimedes also invented a division of a square into 14 pieces leading to a game similar to Tangrams involving making figures from the 14 pieces. Tangrams are of Chinese origin and require little mathematical skill. It is interesting however to see how many convex figures you can make from the 7 tangram pieces. Note again the number 7 which seems to have been associated with magical properties. They were to have a new popularity when Dodgson, writing as Lewis Carroll, introduced Alice type characters.
You can see a picture of the Alice tangrams at THIS LINK.
Fibonacci, already mentioned above, 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 a Journal is devoted to topics related to the sequence. Here is the famous Rabbit Problem.
One of the earliest mentions of Chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth etc. One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 × 3 board (using normal knight's moves).
Magic squares involve using all the numbers 1, 2, 3, ..., $n^{2}$ to fill the squares of an $n \times n$ board so that each row, each column and both main diagonals sum to the same number. They are claimed to go back as far as 2200 BC when the Chinese called them lo-shu. In the early 16th Century Cornelius Agrippa constructed squares for $n$ = 3, 4, 5, 6, 7, 8, 9 which he associated with the seven planets then known (including the Sun and the Moon). Dürer's famous engraving of Melancholia made in 1514 includes a picture of a magic square.
You can see a picture of Melancholia at THIS LINK and of the magic square in it at THIS LINK.
The number of magic squares of a given order is still an unsolved problem. There are 880 squares of size 4 and 275305224 squares of size 5, but the number of larger squares is still unknown.
Durer's square shown above is symmetrical and other conditions were also studied such as the condition that all the diagonals (traced as if the square was on a torus) added to the same number as the row and column sum. Euler studied this type of square known as a pandiagonal square. No pandiagonal square of order $2(2n + 1)$ can exist but they do for all other orders. For $n = 4$ there are 880 magic squares of which 48 are pandiagonal. Veblen in 1908 used matrix methods to study magic squares.
Other early inventors of games included Recorde and Cardan. Cardan invented a game consisting of a number of rings on a bar.
You can see a picture of Cardan's rings at THIS LINK.
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. This problem is similar to the Towers of Hanoi described below. In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
Tartaglia, who with Cardan jointly discovered the algebraic solution of the cubic, was another famous inventor of mathematical recreations. He invented many arithmetical problems, and contributed to problems with weighing masses with the smallest number of weights and Ferry Boat type problems which now have solutions using graph theory.
Bachet was famed as a poet, translator and early mathematician of the French Academy. He is best known for his translation of 1621 of Diophantus's Arithmetica. This is the book which Fermat was reading when he inscribed the margin with his famous Last Theorem. Bachet, however, is also famed as a collector of mathematical puzzles which he published in 1612 Problèmes plaisans et délectables qui font par les nombres Ⓣ. It contains many of the problems referred to above, river crossing problems, weighing problems, number tricks, magic squares etc. Here is an example of one of Bachet's weighing problems
Euler is perhaps the mathematician whose puzzles have led to the most deep mathematical disciplines. In addition to magic square problems and number problems he considered the Knight's Tour of the chess board, the Thirty Six Officers problem and the Seven Bridges of Königsberg.
Euler was not the first to examine the Knight's Tour problem. De Moivre and Montmort had looked at it and solved the problem in the early years of the 18th Century after the question had been posed by Taylor. Ozanam and Montucla quote the solutions of both De Moivre and Montmort. Euler, in 1759 following a suggestion of L Bertrand of Geneva, was the first to make a serious mathematical analysis of it, introducing concepts which were to become important in graph theory. Lagrange also contributed to the understanding of the Knight's Tour problem, as did Vandermonde.
The Seven Bridges of Königsberg heralds the beginning of graph theory and topology.
You can see a picture of the Königsberg bridges at THIS LINK.
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 chess board problem is the Eight queens problem. This problem asks in how many ways 8 queens can be placed on a chess board so that no two attack each other. The generalised problem, in how many ways can $n$ queens be placed on an $n \times n$ board so that no two attack each other, was posed by Franz Nauck in 1850. In 1874 Günther and Glaisher described methods for solving this problem based on determinants. There is a unique solution (up to symmetry) to the 6 × 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for one penny.
In 1857 Hamilton described his Icosian game at a meeting of the British Association in Dublin. It was sold to J. Jacques and Sons, makers of high quality chess sets, for £25 and patented in London in 1859. The game is related to Euler's Knight's Tour problem since, in today's terminology, it asks for a Hamiltonian circuit in a certain graph. The game was a failure and sold very few copies.
You can see a picture of the Icosian game at THIS LINK.
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 two professional inventors of mathematical puzzles, Sam Loyd and Henry Ernest Dudeney, were entertaining the world with a large number of mathematical games and recreations. Loyd's most famous game was the 15 puzzle.
You can see the 15 puzzle at THIS LINK.
It illustrates important properties of permutations.
Loyd was also famous for his chess puzzles. He invented a number of puzzles, some of which are very hard, which he published in the first number of the American Chess Journal.
Édouard Lucas invented the Towers of Hanoi in 1883.
You can see more about this at THIS LINK.
The game of pentominoes is of more recent invention. The problem of tiling an 8 × 8 square with a square hole in the centre was solved in 1935. This problem was shown by computer to have exactly 65 solutions in 1958. In 1953 more general polyominoes were introduced. It is still an unsolved problem how many distinct polyominoes of each order there are. There are 12 pentominoes, 35 hexominoes and 108 heptominoes (including one rather dubious one with a hole in the middle!). Puzzles with polyominoes were invented by Solomon W. Golomb, a mathematician and electrical engineer at Southern California University.
There is a 3-dimensional version of pentominoes where cubes are used as the basic elements instead of squares. A 3 × 4 × 5 rectangular prism can be made from the 3-dimensional pentominoes. Closely related to these is Piet Hein's Soma Cubes. This consists of 7 pieces, 6 pieces consisting of 4 small cubes and one of 3 small cubes. The aim of this game is to assemble a 3 × 3 × 3 cube. This can in fact be done in 230 essentially different ways!
You can see a picture of the Soma Cubes at THIS LINK.
A slightly older game (1921) but still a cube game is due to P A MacMahon and called 30 Coloured Cubes Puzzle. There are 30 cubes which have all possible permutations of precisely 6 colours for their faces. (Can you prove there are exactly 30 such cubes?) Choose a cube at random and then choose 8 other cubes to make a 2 × 2 × 2 cube with the same arrangement of colours for it's faces as the first chosen cube. Each face of the 2 × 2 × 2 cube has to be a single colour and the interior faces have to match in colour.
Raymond Smullyan, a mathematical logician, composed a number of chess problems of a very different type from those usually composed. They are know known as problems of retrograde analysis and their object is to deduce the past history of a game rather than the future of a game which is the conventional problem. Problems of retrograde analysis are problems in mathematical logic.
Here is the first one composed by Smullyan in 1925 when he was 16 years old at THIS LINK.
One of the most important of the modern professional puzzle inventors and collectors is Martin Gardner who wrote an extremely good column in Scientific American for about 30 years, stopping about four years ago. He published some of Smullyan's retrograde analysis chees problems in 1973. He also reported on a computing game in 1973. Of course the advent of personal computers has made both the writing and playing of mathematical games for computers an important new direction. The game Gardner reported on was 'Spirolaterals' devised by Frank Olds with only 3 or 4 lines of code.
You can see some examples of spirolaterals at THIS LINK.
The most famous of recent puzzles in 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 realise 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.
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.
You can see a picture of the Rhind papyrus at THIS LINK.
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 hekats of wheat. What is the total of all of these?Similar problems appear in Fibonacci's Liber Abaci Ⓣ written in 1202 and the familiar St Ives Riddle of the 18th Century based on the same idea (and on the number 7).
Greek mathematics produced many classic puzzles. Perhaps the most famous are from Archimedes in his book The Sandreckoner where he gives the Cattle Problem.
If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun...In some interpretations of the problem the number of cattle turns out to be a number with 206545 digits!
You can see further details at THIS LINK.
Archimedes also invented a division of a square into 14 pieces leading to a game similar to Tangrams involving making figures from the 14 pieces. Tangrams are of Chinese origin and require little mathematical skill. It is interesting however to see how many convex figures you can make from the 7 tangram pieces. Note again the number 7 which seems to have been associated with magical properties. They were to have a new popularity when Dodgson, writing as Lewis Carroll, introduced Alice type characters.
You can see a picture of the Alice tangrams at THIS LINK.
Fibonacci, already mentioned above, 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 a Journal is devoted to topics 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?Fibonacci writes out the first 13 terms of the sequence but does not give the recurrence relation which generates it.
One of the earliest mentions of Chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth etc. One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 × 3 board (using normal knight's moves).
Magic squares involve using all the numbers 1, 2, 3, ..., $n^{2}$ to fill the squares of an $n \times n$ board so that each row, each column and both main diagonals sum to the same number. They are claimed to go back as far as 2200 BC when the Chinese called them lo-shu. In the early 16th Century Cornelius Agrippa constructed squares for $n$ = 3, 4, 5, 6, 7, 8, 9 which he associated with the seven planets then known (including the Sun and the Moon). Dürer's famous engraving of Melancholia made in 1514 includes a picture of a magic square.
You can see a picture of Melancholia at THIS LINK and of the magic square in it at THIS LINK.
The number of magic squares of a given order is still an unsolved problem. There are 880 squares of size 4 and 275305224 squares of size 5, but the number of larger squares is still unknown.
Durer's square shown above is symmetrical and other conditions were also studied such as the condition that all the diagonals (traced as if the square was on a torus) added to the same number as the row and column sum. Euler studied this type of square known as a pandiagonal square. No pandiagonal square of order $2(2n + 1)$ can exist but they do for all other orders. For $n = 4$ there are 880 magic squares of which 48 are pandiagonal. Veblen in 1908 used matrix methods to study magic squares.
Other early inventors of games included Recorde and Cardan. Cardan invented a game consisting of a number of rings on a bar.
You can see a picture of Cardan's rings at THIS LINK.
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. This problem is similar to the Towers of Hanoi described below. In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
Tartaglia, who with Cardan jointly discovered the algebraic solution of the cubic, was another famous inventor of mathematical recreations. He invented many arithmetical problems, and contributed to problems with weighing masses with the smallest number of weights and Ferry Boat type problems which now have solutions using graph theory.
Bachet was famed as a poet, translator and early mathematician of the French Academy. He is best known for his translation of 1621 of Diophantus's Arithmetica. This is the book which Fermat was reading when he inscribed the margin with his famous Last Theorem. Bachet, however, is also famed as a collector of mathematical puzzles which he published in 1612 Problèmes plaisans et délectables qui font par les nombres Ⓣ. It contains many of the problems referred to above, river crossing problems, weighing problems, number tricks, magic squares etc. Here is an example of one of Bachet's weighing problems
What is the least number of weights that can be used on a scale pan to weigh any integral number of pounds from 1 to 40 inclusive, if the weights can be placed in either of the scale pans?You can see the answer at THIS LINK.
Euler is perhaps the mathematician whose puzzles have led to the most deep mathematical disciplines. In addition to magic square problems and number problems he considered the Knight's Tour of the chess board, the Thirty Six Officers problem and the Seven Bridges of Königsberg.
Euler was not the first to examine the Knight's Tour problem. De Moivre and Montmort had looked at it and solved the problem in the early years of the 18th Century after the question had been posed by Taylor. Ozanam and Montucla quote the solutions of both De Moivre and Montmort. Euler, in 1759 following a suggestion of L Bertrand of Geneva, was the first to make a serious mathematical analysis of it, introducing concepts which were to become important in graph theory. Lagrange also contributed to the understanding of the Knight's Tour problem, as did Vandermonde.
The Seven Bridges of Königsberg heralds the beginning of graph theory and topology.
You can see a picture of the Königsberg bridges at THIS LINK.
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 chess board problem is the Eight queens problem. This problem asks in how many ways 8 queens can be placed on a chess board so that no two attack each other. The generalised problem, in how many ways can $n$ queens be placed on an $n \times n$ board so that no two attack each other, was posed by Franz Nauck in 1850. In 1874 Günther and Glaisher described methods for solving this problem based on determinants. There is a unique solution (up to symmetry) to the 6 × 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for one penny.
In 1857 Hamilton described his Icosian game at a meeting of the British Association in Dublin. It was sold to J. Jacques and Sons, makers of high quality chess sets, for £25 and patented in London in 1859. The game is related to Euler's Knight's Tour problem since, in today's terminology, it asks for a Hamiltonian circuit in a certain graph. The game was a failure and sold very few copies.
You can see a picture of the Icosian game at THIS LINK.
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 two professional inventors of mathematical puzzles, Sam Loyd and Henry Ernest Dudeney, were entertaining the world with a large number of mathematical games and recreations. Loyd's most famous game was the 15 puzzle.
You can see the 15 puzzle at THIS LINK.
It illustrates important properties of permutations.
Loyd was also famous for his chess puzzles. He invented a number of puzzles, some of which are very hard, which he published in the first number of the American Chess Journal.
Édouard Lucas invented the Towers of Hanoi in 1883.
You can see more about this at THIS LINK.
The game of pentominoes is of more recent invention. The problem of tiling an 8 × 8 square with a square hole in the centre was solved in 1935. This problem was shown by computer to have exactly 65 solutions in 1958. In 1953 more general polyominoes were introduced. It is still an unsolved problem how many distinct polyominoes of each order there are. There are 12 pentominoes, 35 hexominoes and 108 heptominoes (including one rather dubious one with a hole in the middle!). Puzzles with polyominoes were invented by Solomon W. Golomb, a mathematician and electrical engineer at Southern California University.
There is a 3-dimensional version of pentominoes where cubes are used as the basic elements instead of squares. A 3 × 4 × 5 rectangular prism can be made from the 3-dimensional pentominoes. Closely related to these is Piet Hein's Soma Cubes. This consists of 7 pieces, 6 pieces consisting of 4 small cubes and one of 3 small cubes. The aim of this game is to assemble a 3 × 3 × 3 cube. This can in fact be done in 230 essentially different ways!
You can see a picture of the Soma Cubes at THIS LINK.
A slightly older game (1921) but still a cube game is due to P A MacMahon and called 30 Coloured Cubes Puzzle. There are 30 cubes which have all possible permutations of precisely 6 colours for their faces. (Can you prove there are exactly 30 such cubes?) Choose a cube at random and then choose 8 other cubes to make a 2 × 2 × 2 cube with the same arrangement of colours for it's faces as the first chosen cube. Each face of the 2 × 2 × 2 cube has to be a single colour and the interior faces have to match in colour.
Raymond Smullyan, a mathematical logician, composed a number of chess problems of a very different type from those usually composed. They are know known as problems of retrograde analysis and their object is to deduce the past history of a game rather than the future of a game which is the conventional problem. Problems of retrograde analysis are problems in mathematical logic.
Here is the first one composed by Smullyan in 1925 when he was 16 years old at THIS LINK.
One of the most important of the modern professional puzzle inventors and collectors is Martin Gardner who wrote an extremely good column in Scientific American for about 30 years, stopping about four years ago. He published some of Smullyan's retrograde analysis chees problems in 1973. He also reported on a computing game in 1973. Of course the advent of personal computers has made both the writing and playing of mathematical games for computers an important new direction. The game Gardner reported on was 'Spirolaterals' devised by Frank Olds with only 3 or 4 lines of code.
You can see some examples of spirolaterals at THIS LINK.
The most famous of recent puzzles in 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 realise 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.
References (show)
- L Bachelier, Le jeu, la chance et le hasard : Reprint of the 1914 original (Sceaux, 1993).
- W W Rouse Ball, Mathematical Recreations and Essays (London, 1940).
- W W Rouse Ball and H S M Coxeter, Mathematical recreations and essays (New York, 1987).
- A Carrer, Elementary study of a most ancient Mesopotamian game (Italian), Atti Accad. Sci. Istit. Bologna Cl. Sci. Fis. Rend. (13) 7 (1) (1979/80), 97-121.
- R W Dimand and M A Dimand, The early history of the theory of strategic games from Waldegrave to Borel, in Toward a history of game theory (Durham, NC, 1992), 15-27.
- C L Dodgson, Mathematical recreations of Lewis Carroll. Vol. 1 : Symbolic logic and The game of logic (New York, 1958).
- C L Dodgson, Mathematical recreations of Lewis Carroll. Vol. 2 : Pillow problems and A tangled tale (New York, 1958).
- H E Eiss, Dictionary of mathematical games, puzzles, and amusements (Westport, CT, 1988).
- M Kraitchik, Mathematical recreations (New York, N. Y., 1953).
- E Lucas, Récréations mathématiques (Paris, 1960).
- O Ore, Cardano, the gambling scholar (Princeton, N. J., 1953).
- I Schneider, The marketplace and games of chance in the fifteenth and sixteenth centuries, Mathematics from manuscript to print, 1300-1600 (New York, 1988), 220-235.
- V Thébault, Les récréations mathématiques (parmi les nombres curieux) : Avec des notes de A. Buquet (Paris, 1952).
Additional Resources (show)
Other pages about Mathematical games:
Written by
J J O'Connor and E F Robertson
Last Update May 1996
Last Update May 1996