An Historical Background to Some Mathematical GamesAuthor(s): Michael CorneliusSource: Mathematics in School, Vol. 15, No. 1 (Jan., 1986), pp. 47-49Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216313 .

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Player A

The Wari Board Player B

An HistOrical Background

to some

Mathemalical Games by Michael Cornelius,

University of Durham

Puzzles, games and recreations in mathematics have been in existence for a very long time. Most items in modern collections have their origins in problems posed many years (usually hundreds of years) ago. Perusal of accounts of the history of mathematics or the history of games can often lead to rewarding discoveries and, at a time when teachers of mathematics are being exhorted to initiate pupil investig- ations, here are rich sources of ideas and material. There is evidence, in particular from Egypt and China, of mathemat- ical games dating back as far as 2000 BC. Games have emerged from all parts of the world - some have achieved universal popularity, others have remained in particular regions. It is impossible in a short account to give a detailed discussion of the history of games and recreations, what follows is an attempt to look at a few interesting examples which will provide no more than snapshots but it is hoped that they will encourage teachers to explore further and experiment with their pupils.

Wari Wari is an example of a mancala game i.e. a game played on a board with rows of cup-shaped holes which hold a number of counters. It has been played in Egypt for thousands of years and was a pastime of African slaves. The Wari board consists of two rows of cups, each cup initially containing four counters (beads, nuts, dry beans etc):

A and B play alternately, lifting the "seeds" from one of the holes on their side of the board and sowing one into each hole in an anticlockwise direction.

If the last seed drops into an enemy hole to make a final

total of 2 or 3 the seeds are captured and the seeds on the opponent's side contiguous with and behind the plundered hole are also taken. The player holding most seeds at the end is the winner.

As an example, we could denote the starting position by A 4 4 4 4 4 4 B 4 4 4 4 4 4

which after one move by B could become

A 4 4 4 4 4 5 B 4 4 0 5 5 5

If at some stage in a game the position was

A 1 2 1 1 2 2 B 4 4 2 7 1 0

and it was B's turn to play, sowing the 7 would leave

A 132233 B 4 4 2 0 2 1

and B would capture 3 + 2 + 2 + 3 + 3= 13 seeds. A cup may become heavily loaded during a game and

when emptied it may need more than one cycle of the board - in this case the emptied cup is omitted from the sowing on the second and subsequent cycles.

Teachers may like to think of Wari as an interesting investigational exercise, (what is the "best" first move? What are the best tactics?) perhaps combining with some historical background or even the construction of a Wari board in a woodwork lesson.

Mathematics in School, January 1986 47

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Nine Men's Morris This is one of the oldest board games. Boards have been found carved on a roofing slab from an Egyptian temple (c. 1400 BC) and on a stone from a Bronze Age burial site in Ireland.

Two players (black and white) each have nine counters to place at any of the 24 points on the board. In part one, players play alternately placing men on the board, in part two a turn is taken by moving a piece to an adjacent vacant point. The object is to form a "mill" or row of three - each time this is done, a player may remove one of the opponent's men. The winner either blocks all his opponent's men or reduces their number to two. A simpler version, "Three Men's Morris" is played on a board with 9 points with players having four counters each:

here the aim is simply to get three men in a line. In the cathedrals of Norwich, Canterbury, Gloucester, Salisbury and Westminster Abbey there are boards cut into the cloister seats doubtless to relieve the tedium of long services!

The game cries out for a classroom investigation involv- ing geometry (?) algebra (?) ..

Fox and Geese A "hunt" game said to be enjoyed by Queen Victoria and going back some 700 years to origins in Iceland, England and Italy. It is played on a "solitaire" board:

GGG

GGG GGGGGGG Go oGoGoG o G

Go o F o o G

0oo

0 0

The fox (F) must try to kill seventeen geese (G) and has first move. It moves forward, backward or diagonally or to the side and "kills" a victim by jumping over it into a vacant hole. A dead goose is removed from the board. The geese may move only forward or sideways and must attempt to corner the fox so that it cannot move.

Who should win? What about different numbers of geese (originally the game had only 13)? More investigation!

Hopscotch If some of these games seem more appropriate for older pupils, surely hopscotch has younger appeal. Its origins are unclear but there is a hopscotch diagram on the floor of the Forum in Rome and it is played widely in Great Britain, Russia, India and China - the "scotch" comes from a word meaning to mark or score lightly as children would with a stick on the ground. Most readers will be familiar with the "game". Two versions of the diagram may be of interest:

8 9

7

5 6

4

3

2

1

5 6 MOON

4 7

3 8

2 9

1 10

EARTH

In the "Moon" version a minus sign indicates hopping on one foot, a plus sign on both feet. It is surely not difficult to see possibilities for some geometry of shapes, simple num- ber work and even use of signs.

Ko-No, Mu-Torere and Tick-Tack-Toe These are three simple games which could provide some basic analysis for pupils. Ko-no is played by children in China and India:

A IB

X Y One player has counters at A, B; the other at X, Y. A

move must be to the vacant space - a player loses when he is blocked. Teachers may be surprised at the problem solving skills and strategies which this simple game de- velops in young children.

48 Mathematics in School, January 1986

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Mu-torere is from the Maoris in New Zealand:

3

2 4

Player A starts with pieces on 1, 2, 3, 4; player B on 5, 6, 7, 8. A man may move either to the central point or to the end of an adjacent ray (if empty). The first player to blockade his opponent wins (the first move must allow the second player to play).

An example of a game:

1. 5-9 4-5 2. 9-4 3-9 3. 4-3 9-4 4. 3-9 2-3 5. 9-2 4-9 winning

Tick-Tack-Toe (Noughts and Crosses) will be familiar to all readers. It appears to have its origins in England, Holland and Sweden but its date is uncertain. When a player succeeds in making a row he exclaims:

"Tit tat toe, Here I go, Three jolly butcher boys all in a row".

There are only three possible first moves (can pupils work this out?) and twelve positions after the second player has moved. Should the first player always win? (An OHP transparency might help illustrate the limited possibilities by using rotations etc.) The game was frequently played when pupils had slates - much better than pencil and paper!

Magic Squares Magic squares were constructed in China as early as 2000 BC, they appear to have been introduced into Europe in the fifteenth century and a 4 x 4 square was included in Albrecht Durer's "Meloncholia" engraved in 1514. An easy way of arriving at a 3 x 3 square is:

1

4 2

7 5 3

8 6

9

4 9 2

3 5 7

8 1 6

and the method will extend to other odd order squares. A number of games can be developed e.g. for the 3 x 3

square, one player takes pieces numbered 1, 3, 5, 7, 9 and the other 2, 4, 6, 8. The "odd" player begins placing a piece anywhere on the grid, the opponent follows and the object is to make a row of three numbers adding up to 15.

Mathematical Recreations Puzzles and recreations in modern collections often have their origins in the history of mathematics, some of the following will be familiar to most readers but the back-

ground may not be known. 1. A wolf, a goat and a cabbage must be moved across a river in a boat holding only one besides the ferryman. How must he carry them across so that the goat shall not eat the cabbage, nor the wolf the goat? From: Problems for the Quickening of the Mind by Alcuin of York (c. 775). How many versions of this problem have since appeared in "modern" guises? 2. What is the least number of weights needed to weigh any whole number of pounds from 1 lb to 40 lb? (Using either one scale pan or both). Recent textbooks have often used this problem for work in bases two and three. A solution was given by Tartaglia in 1556 and the problem appears in a famous collection by Bachet in "Problemes plaisans et delectables" (1612). 3. Euler in 1782 posed the problem: "Is it possible to arrange 36 officers each having one of 6 different ranks and each belonging to one of 6 different regiments in a square 6 x 6 formation so that each row and file contains just one officer of each rank and one from each regiment?" 4. The "Hamiltonian Game" (1857) named after Sir Will- iam Hamilton denotes the twenty vertices of a regular dodecahedron by letters representing towns. The problem is to visit every town once and return to the starting town. A possible solution is AB CD E F G H J K L M N O P Q R S T U - how many solutions are there?

Q R

M L ~K

C) E

B O

pC~ ~ ~H~S

A\ /G

U

T

5. In similar vein, Kirkman (1847) proposed the problem: "A school mistress takes 15 girls for a daily walk. They go in 5 rows of 3. Arrange them so that for 7 consecutive days no girl will walk with the same partner more than once".

Delving into the history of mathematics and the history of games can be a fascinating pastime and can yield some entertaining, amusing and useful pieces of mathematics. In the spirit of current "Problem Solving" and "Investig- ations" perhaps this brief collection of puzzles and games will be sufficient to spark some interest and provide some ideas for possible use by teachers in the classroom.

In producing this miscellany of games and puzzles, I have drawn freely on the sources listed in the bibliography below, all are likely to provide some fascinating browsing and a wealth of ideas.

References 1. Ball, W. W. M. and Coxeter, H. S. M. (1974) Mathematical Recreations

and Essays, Univ. of Toronto Press, 12th Edition. 2. Bell, R. C. (1960) Board and Table Games, Oxford Univ. Press. 3. Eves, H. (1965) An Introduction to the History of Mathematics, Holt,

Rinehart and Winston. 4. Grunfeld, F. V. (1975) Games of the World, Holt, Rinehart and

Winston. 5. McConville, R. (1974) The History of Board Games, Creative Public-

ations Inc. 6. Murray, H. J. R. (1978) A History of Board Games other than Chess,

Hacker Art Books Inc.

Mathematics in School, January 1986 49

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