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.

 2 2 2 2 0 2 2 2 2

 15 12 12 15 12 18 18 12 15 12 12 15

 140 128 86 128 140 120 144 108 144 120 140 128 86 128 140

 833 700 582 582 700 833 804 804 574 574 804 804 833 700 582 582 700 833

 4643 3982 2913 3332 2913 3982 4643 4294 4556 3242 2932 3242 4556 4294 4643 3982 2913 3332 2913 3982 4643

 27824 23886 18304 18375 18375 18304 23886 27824 26292 26724 19038 17966 17966 19038 26724 26292 27824 23886 18304 18375 18375 18304 23886 27824

 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

 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

 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

 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