The Distribution of the Knight

Dr Colin Rose
 

Number of Paths:   3 × m

For any given starting point, how many paths exist (knight tours + dead ends)? This page displays the answer for all (3 × m) chess boards, for m ≤ 12.

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.

This probability may takes values from 0 to 1/3.

3 × 3
2 2 2
2 0 2
2 2 2


3 × 4
15 12 12 15
12 18 18 12
15 12 12 15


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


3 × 6
833 700 582 582 700 833
804 804 574 574 804 804
833 700 582 582 700 833


3 × 7
4643 3982 2913 3332 2913 3982 4643
4294 4556 3242 2932 3242 4556 4294
4643 3982 2913 3332 2913 3982 4643


3 × 8
27824 23886 18304 18375 18375 18304 23886 27824
26292 26724 19038 17966 17966 19038 26724 26292
27824 23886 18304 18375 18375 18304 23886 27824


3 × 9
165155 140354 107958 113359 103150 113359 107958 140354 165155
157668 161226 114332 106444 109776 106444 114332 161226 157668
165155 140354 107958 113359 103150 113359 107958 140354 165155


3 × 10
968532 819862 629021 656397 627555 627555 656397 629021 819862 968532
927766 942486 656114 631118 638592 638592 631118 656114 942486 927766
968532 819862 629021 656397 627555 627555 656397 629021 819862 968532


3 × 11
5611265 4740822 3640058 3750102 3594531 3738552 3594531 3750102 3640058 4740822 5611265
5390962 5438448 3774276 3588074 3725556 3655456 3725556 3588074 3774276 5438448 5390962
5611265 4740822 3640058 3750102 3594531 3738552 3594531 3750102 3640058 4740822 5611265


3 × 12
32397038 27325918 20984726 21576046 20503421 21222770 21222770 20503421 21576046 20984726 27325918 32397038
31223338 31394926 21690708 20638264 21129754 21164064 21164064 21129754 20638264 21690708 31394926 31223338
32397038 27325918 20984726 21576046 20503421 21222770 21222770 20503421 21576046 20984726 27325918 32397038