A difficult game

The following rather difficult two person game is based on combinatorics but neatly illustrates a number of useful design principles:

1.

Begin with the set of non-zero digits, {1, ..., 9}.

2.

The two players take turns selecting a digit from the set.

3.

Once selected, a digit is removed from the set of available digits making it unavailable for future selections by either player.

4.

The first player to have three digits which sum to 15 is the winner.

5.

The game ends in a draw if all of the digits have been selected and neither player has three digits summing to 15.

Imagine playing the game. The goal of the game is simply defined. The moves are relatively straightforward. Yet the game is difficult to play. Why?

Several factors contribute to the difficulty. To begin with, the game is introduced and described in such a way that one has the perception that it is difficult even before playing it. The gratuitous mathematical¹ description especially heightens this impression for non-mathematical readers.

Playing the game reinforces this perception. At each turn at least three sets of information need to be managed: your set of selected digits, your opponent's, and those remaining. Each turn requires a choice between several competing options. Each option needs to be explored one or two moves further out in order to assess its merits.

The slow, serial nature of conscious thought and the known limitations of short term memory make management of this complexity difficult for most people. We need to introduce some tricks to help us out.

The first thing most people will do is write the set of numbers down and record the selections of both players. Next, writing down all possible triples of digits summing to 15 helps in determining the possible outcomes of different moves.

Further analysis of the possible triples suggests the following remarkable tabular arrangement (from Norman, 1988, page 126):

8		1		6
3		5		7
4		9		2

Not only do the three rows, three columns, and two diagonals each sum to 15 but they actually exhaust the set of triples which do! Moves can be identified directly on the display by, say, circling our selections and crossing out those of our opponent. The digits are now superfluous and the object of the game is to get three circles (or crosses) in a straight line - tic-tac-toe!

By building the structure of the game into the display, the complexity is substantially reduced and the players are free to concentrate more on the play and on developing winning strategies.

Tic-tac-toe is an example of a wide and deep structure - many choices at each turn, many turns in sequence. Figure

  

Figure: Tic-tac-toe: A wide and deep structure.

shows the essential structure of the game when the first player (O) selects the middle square on the first turn. From this initial move, player one can now follow a strategy which ensures that the game ends in either a draw or a win.