Many important ideas can be taught through games.  This first article in a series describes a game which can be fun as well as instructive.

The flavour of fudge depends, among other things, on the proportion of butter to sugar.  If you were to make several pots of fudge with different amounts of butter and sugar, you may be able to decide which particular mix had the best flavour.

There are many practical problems in industry like this, where the combination of values of several variables must be found to give the best value of some property of a product or process.  Another simple example may be a chemical process whose yield of a product depends on the temperature and pressure in the reaction vessel.

Looking for the best value of a property according to all possible values of two variables is rather like trying to guess the highest point on a map by stating the grid references.  The subject of statistics gives us some techniques for tackling such problems.  They include design of sequential experiments and estimation of trends, or slopes, even when there is error in the property being measured.  In this article I describe a game which may help to introduce some of these ideas to students.  It is rather like the game of battleships.

Battleships was played by schoolchildren certainly 40 years ago and is still played today with gusto.  It has the advantage that it can be played anywhere between two people with scraps of paper and pencils. I have known seven-year-olds find it fun, yet adults play it with just as much relish plus a tease of logic.

Blindfold climbing is similar to battleships.  Not such young children can play it, but ten-year-olds have succeeded with both understanding and pleasure.  They have learned to create, as well as recognise, map grid references and height contours.  For older children and adults, the game is an excellent introduction to experimental hill climbing. I claim no credit for the game; versions of it have been suggested by various people. I have, however, used it repeatedly to illustrate some ideas of experimental design to metallurgists, chemists, and engineers. I have also taught it to my children as a game to play on rainy days.

There are two players.  Each has a sheet of paper on which he draws two rectangles, fifteen centimetres wide by ten centimetres deep.  Each rectangle is marked faintly with a one-centimetre grid.  Standard graph paper may be used.  The grids are marked with a scale of one centimetre representing ten metres.  Each player marks his rectangles mine and his, then on the one marked mine he draws a set of contours with a unique highest point.  The contours should be marked with 100 metre intervals and the highest point should be between 500 and 1000 metres above the lowest point.  A little fun is added by drawing the map as a treasure island with the chest buried at the peak.

The players hide their charts from each other and take it in turn to call a grid reference, or pair of co-ordinates, to which the other responds with the height read from his own map.  The questioner marks the height at that grid reference and tries to reconstruct his opponent's contours.  The game ends as soon as either player reaches within five metres, horizontally, of his opponent's peak.  Alternatively players may pay a token for each height given and stop when they like.  The one who reaches closest to the peak for the least cost is the winner.  Thus the idea of costs of experiments may be introduced.  More than two people can play, but then the rules have to be adapted.  One possibility is to have one of the players act as God and the others race to his mountain top.  The game is also an excellent means for illustrating some ideas of experimental design to a class. I put up a partition on a stage in front of the audience.  On either side of the partition there is a blackboard large enough to draw two rectangular grids.  Perhaps a well equipped modern classroom will provide an overhead projector on either side of the partition.  The grids can be drawn on acetate with permanent ink and the game played with washable ink, to save the trouble of redrawing grids between games.

When the game has been described to the class, two players take their positions where they cannot see each other but the spectators can see both of them.

If the game is presented well, the class will join with the competing spirit, siding with their favourite and even placing bets.  The players act their roles: the loser is seen to suffer and the winner is seen to glow.

But the game is fixed!

One player will have agreed in advance to follow a strategy by which he will probably, though not certainly, lose.  One such strategy is to choose his grid points randomly, or haphazardly.  Another is to follow what is known as the classical research method: keeping one variable fixed at a single value while testing equally spaced values of the other variable; then keeping the second variable fixed at the highest point found, while testing the first variable at equally spaced values.

Meanwhile, the other player will follow a strategy by which he will probably, though again not certainly, win.  Such a strategy will be one of the standard and well proven evolutionary operation procedures (EVOP) for hill climbing on experimental response surfaces.  One of these is the simplex which, for the two variable case, can be defined as follows:

Step one: Test three points at the vertices of an equilateral triangle.
Step two: Determine the lowest of the three points and then test a fourth point which is at the reflection of the lowest point through the other two.
Step three:  Reject the first lowest point (leaving only three points) and return to step two.

If at some stage you return to an earlier set of points, you are on a ridge and must choose the second lowest to escape from it.  There are more rules about the simplex than these.  They are for dealing with more than two dimensions and difficult situations.  These will be left for another article.

Another strategy for hill climbing is to use the 2 x 2 factorial which, in its simplest form, is:

Step one:     Test four points at the vertices of a square.
Step two:      Estimate the direction of greatest slope and return to step one.

Trials can be saved by using some points already tested as vertices of later squares.

The game is clearly useful for introducing many ideas at several levels of thought.  Doubtless more variations of the game will be discovered and more winning and losing strategies may be illustrated. I should like to hear of any.

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