Fig. 2. Four directions.
What are the 4 operations with numbers which Simon can carry out? Let us consider some simple operations that we can perform with just 4 numbers. Suppose that they stood for 4 directions in the order east, north, west, south ([see Fig. 2]). Or suppose that they stood for a turn counterclockwise through some right angles as follows:
- 0: Turn through 0°, or no right angles.
- 1: Turn through 90°, or 1 right angle.
- 2: Turn through 180°, or 2 right angles.
- 3: Turn through 270°, or 3 right angles.
Then we could have the operations of addition and negation, defined as follows:
| Addition | Negation | |||||||
|---|---|---|---|---|---|---|---|---|
| c = a + b | c = -a | |||||||
| b: | 0 | 1 | 2 | 3 | ||||
| a: | a | c | ||||||
| 0 | 0 | 1 | 2 | 3 | 0 | 0 | ||
| 1 | 1 | 2 | 3 | 0 | 1 | 3 | ||
| 2 | 2 | 3 | 0 | 1 | 2 | 2 | ||
| 3 | 3 | 0 | 1 | 2 | 3 | 1 | ||
For example, the first table says, “1 plus 3 equals 0.” This means that, if we turn 1 right angle and then turn in the same direction 3 more right angles, we face in exactly the same way as we did at the start. This statement is clearly true. For another example, the second table says, “2 is the negative of 2.” This means that, if we turn to the left 2 right angles, we face in exactly the same way as if we turn to the right 2 right angles, and this statement also is, of course, true.
With only these two operations in Simon, we should probably find him a little too dull to tell us much. Let us, therefore, put into Simon two more operations. Let us choose two operations involving both numbers and logic: in particular, (1) finding which of two numbers is greater and (2) selecting. In this way we shall make Simon a little cleverer.
It is easy to teach Simon how to find which of two numbers is the greater when all the numbers that Simon has to know are 0, 1, 2, 3. We put all possible cases of two numbers a and b into a table:
| b: | 0 | 1 | 2 | 3 | ||||
| a: | ||||||||
| 0 | ||||||||
| 1 | ||||||||
| 2 | ||||||||
| 3 | ||||||||