## The Distribution of the Knight

Dr Colin Rose

### Number of Paths:   4 × m

For any given starting point, how many paths exist (knight tours + dead ends)? This page displays the answer for all (4 × m) chess boards, for m < 9.

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.

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

 760 668 668 760 668 562 562 668 668 562 562 668 760 668 668 760

 12856 13085 10112 13085 12856 13091 10178 6338 10178 13091 13091 10178 6338 10178 13091 12856 13085 10112 13085 12856

 234481 227671 183417 183417 227671 234481 226103 192072 110957 110957 192072 226103 226103 192072 110957 110957 192072 226103 234481 227671 183417 183417 227671 234481

 4767809 4519316 3740155 3775750 3740155 4519316 4767809 4459714 3836066 2361451 2212492 2361451 3836066 4459714 4459714 3836066 2361451 2212492 2361451 3836066 4459714 4767809 4519316 3740155 3775750 3740155 4519316 4767809

 94453126 89817510 73496915 76391894 76391894 73496915 89817510 94453126 88336320 75080733 46295469 47067619 47067619 46295469 75080733 88336320 88336320 75080733 46295469 47067619 47067619 46295469 75080733 88336320 94453126 89817510 73496915 76391894 76391894 73496915 89817510 94453126