The Distribution of the Knight

Dr Colin Rose
 

Number of Paths:   5 × m

For any given starting point, how many paths exist (knight tours + dead ends)? This page displays the answer for all (5 × m) chess boards, for m < 8.

For any given square, we can now find the probability that a randomly chosen path is a Knight's Tour. If X = No. of Knight's Tours, and Y = No. of Paths, then

Prob[ random path is Knight's Tour ] = X / Y.

5 × 3
140 120 140
128 144 128
86 108 86
128 144 128
140 120 140


5 × 4
12856 13091 13091 12856
13085 10178 10178 13085
10112 6338 6338 10112
13085 10178 10178 13085
12856 13091 13091 12856


5 × 5
625308 727156 595892 727156 625308
727156 601036 384804 601036 727156
595892 384804 254400 384804 595892
727156 601036 384804 601036 727156
625308 727156 595892 727156 625308


5 × 6
35798626 38275155 33991411 33991411 38275155 35798626
40769573 34283068 21828366 21828366 34283068 40769573
32628352 22412002 14299730 14299730 22412002 32628352
40769573 34283068 21828366 21828366 34283068 40769573
35798626 38275155 33991411 33991411 38275155 35798626


5 × 7
2240456924 2350426879 2051825358 2136903238 2051825358 2350426879 2240456924
2420344251 2046249807 1387283117 1412027680 1387283117 2046249807 2420344251
1947455492 1364268950 915277280 925098592 915277280 1364268950 1947455492
2420344251 2046249807 1387283117 1412027680 1387283117 2046249807 2420344251
2240456924 2350426879 2051825358 2136903238 2051825358 2350426879 2240456924