OPTIMAL
KNIGHT COVER
SOLUTIONS

This page contains complete enumerations of solutions.

All values are optimal and have been verified using exact methods.

The "Knights" column contains the minimum number of knights required for a cover.
(OEIS A006075)

The "RR Solutions" column contains the number of solutions,
counting rotations and reflections as the same.
(OEIS A006076)

The "Total Solutions" column contains the number of solutions,
counting rotations and reflections as different.
(OEIS A103315)

Size Knights RR Solutions Total Solutions 1x1 1 1 1 2x2 4 1 1 3x3 4 2 8 4x4 4 3 9 5x5 5 8 47 6x6 8 23 127 7x7 10 3 10 8x8 12 1 2 9x9 14 1 2 10x10 16 2 4

Size Knights RR Solutions Total Solutions 11x11 21 100 800 12x12 24 1 2 13x13 28 20 152 14x14 32 1 4 15x15 36 63 504 16x16 40 1 2 17x17 46 29 212 18x18 52 2551 19,562 19x19 57 19 152 20x20 62 ? ?

Note that there are 8 rotations and reflections of a solution,
but not all of them can produce a difference because
a solution can have its own symmetry.
For example, the lone 12x12 solution has its own 4-way symmetry
producing only 2 total solutions.
As a contrast, all of the 100 11x11 solutions are asymmetrical,
so all 8 rotations and reflections are different,
producing 800 total solutions.