Transformations

Transformations

The 2201 possible playing positions facing the MENACE machine in the first four moves can be reduced to just 304 unique positions by applying the following transformations.

0	1	2
3	4	5
6	7	8

6	3	0
7	4	1
8	5	2

2	5	8
1	4	7
0	3	6

8	7	6
5	4	3
2	1	0

2	1	0
5	4	3
8	7	6

6	7	8
3	4	5
0	1	2

2	1	6
5	4	3
8	7	0

8	5	2
7	4	1
6	3	0

0	3	6
1	4	7
2	5	8


					I have numbered the playing cells zero through 8 (we programmers love to count from zero). This is the normal layout of the board. The eight variations below are numbered to show where the original cell position ends up when applying the relevant transformation

					Here the board has been rotated 90 degrees right

		Here the board has been rotated 90 degrees left

		This is a 180 degree rotation.

	Here the grid has been reflected across the vertical axis (left swapped with right)

				Here the grid has been reflected across the horizontal axis (top swapped with bottom)

		This one results from a 180 degree rotation and then a reflection across the horizontal axis

			This is a 90 degree rotation left and a reflection in the vertical axis

	The last is a 90 degree rotation to the right and a reflection in the vertical axis


	All this means is that a playing position is logically the same in terms of the next best move irrespective of the transformation applied. Donald Michie used the symmetry of the playing grid to reduce his MENACE machine to manageable proportions. We have done the same in an attempt to simulate his machine properly.

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	The first board offers the same playing choice as the second despite the transformation


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