Mathematical_Recreations-Kraitchik-2e

Games 273
U- 2 V-1 + u- 2 v + U-1V-2 + u-1 v2 + Uv-2 + uv2 + U 2V-l + u 2v,
and the results of successive moves are found by expanding
successively higher powers of this expression. However,
in order to be certain that we always get the shortest possi-
1 7 28 77 161 266 357 393 357 266 161 77 28 7 1
7 1 6 21 50 90 126 141 126 90 50 21 6 1 7
28 6 1 5 15 30 45 51 45 30 15 5 1 6 28
77 21 5 1 4 10 16 19 16 10 4 1 5 21 77
161 50 15 4 1 3 6 7 6 3 1 4 15 50 161
266 90 30 10 3 1 2 3 2 1 3 10 30 90 266
357 126 45 16 6 2 1 1 1 2 6 16 45 126 357
393 141 51 19 7 3 1 1 1 3 7 19 51 141 393
357 126 45 16 6 2 1 1 1 2 6 16 45 126 357
266 90 30 10 3 1 2 3 2 1 3 10 30 90 266
161 50 15 4 1 3 6 7 6 3 1 4 15 50 161
77 21 5 1 4 10 16 19 16 10 4 1 5 21 77
28 6 1 5 15 30 45 51 45 30 15 5 1 6 28
7 1 6 21 50 90 126 141 126 90 50 21 6 1 7
7 28 77 161 266 357 393 357 266 161 77 28 7 1
FIGURE 146.
Board.
Algebra of the King's Move, in Chess: Unlimited
ble path we must eliminate at each step any term that has
already occurred. For example, UV2'U-1V-2 = 1, which means
that the knight has gone back to his original position, so this
term must be omitted in forming the second power.
While this algebraic procedure may be used for proving
general propositions about the knight's moves, it is too cumbersome
to be practical for ordinary purposes. Instead it is