Suppose we have a solution that covers the board.

x o x        Case 1: no Figure 1 x-square is occupied in the solution.
o o o        No o-square can be covered by an attack, therefore all o-squares
x o x        must be occupied, and the solution must have at least 5 knights.
Figure 1

x o o        Case 2: a Figure 1 x-square is occupied in the solution.
- o -        Pick one of them, then rotate the board so that its location becomes
- - -        the x-square in Figure 2.  Then the 3 o-squares must each be covered
Figure 2     by a separate knight, so the solution must have at least 4 knights.

- - - a   A single knight cannot cover all 3 a-squares or all 3 b-squares,
a - - b   thus both sets require 2 knights.  Any knight that covers the squares
b - - a   of one letter group does not cover any square of the other letter group.
- - - b   Therefore, 2 separate sets of 2 knights are required to cover them all.

Suppose we have a solution that covers the board.

x x x x x   Case 1: no Figure 1 x-square is occupied in the solution.
x x o x x   Then, similarly to Case 1 of the 3x3 proof, all o-squares
x o o o x   must be covered by occupation, thus the solution
x x o x x   must contain at least 5 knights.
x x x x x
Figure 1

- x - - x   Case 2: a Figure 1 x-square is occupied in the solution.
- x - - a   Pick one of them and rotate the board so that its location
x a - - b   becomes a Figure 2 x-square.  That knight does not contribute
- b - - a   to the cover of the pattern of a-squares and b-squares.
- x - - b   This pattern requires 4 knights as shown in the 4x4 proof.
Figure 2    Thus the solution must contain at least 5 knights.

a b a b a b   It takes 2 knights to cover each letter group.
- - - - - -   A knight that covers the squares of one letter group
- - - - - -   does not cover any squares of the other groups.
- - - - - -   Therefore, it takes 4 separate sets of 2 knights to cover them all.
- - - - - -
c d c d c d

o - - - - - o   A single knight cannot cover more than one o-square,
o o - - - - o   thus it takes 10 knights to cover all 10 o-squares.
- - - - - - -
- - - - - - -
- - - - - - -
- - - - - o o
o o - - - - o

o o - - - - - o   A single knight cannot cover more than one o-square,
- o - - - - o o   thus it takes 12 knights to cover all 12 o-squares.
- - - - - - - -
- - - - - - - -
- - - - - - - -
- - - - - - - -
o o - - - - o -
o - - - - - o o

a b - -   Consider this pattern of 7 a,b,c squares.
b c x -   If the x-square is not occupied, then no knight can cover more than
a b - -   2 a,b,c squares, so it takes 4 knights to cover them all.
- a - -   If the x-square is occupied, then it covers all the a-squares,
- - - -   but the 3 b-squares will require 2 knights and
- - - -   the c-square will require a 3rd separate knight.
          So in both cases, it takes at least 4 knights to cover the pattern.


- - o o o   Consider this pattern of 5 o-squares.
- - o - o   A single knight can cover at most 2 of these squares,
- - - - -   so it takes 3 knights to cover them all.
- - - - -


a a - - - - b b b   In this 9x9 board, there are two instances of each
a a - - - - b - b   of the above patterns.  They are separated so that
a a - - - - - - -   no knight can cover the squares of more than one pattern,
- a - - - - - - -   thus it takes a separate set of knights to cover each.
- - - - - - - - -   Two of the patterns require 4 knights each
- - - - - - - d -   and the other two patterns require 3 knights each.
- - - - - - - d d   This totals 14 knights to cover all patterns.
c - c - - - - d d
c c c - - - - d d

a a - - - - - b b b   Each of the four patterns are separated so that a
a a - - - - b b b b   single knight cannot cover the squares of more than
a a - - - - - - - -   one pattern.  Each pattern requires 4 knights
- a - - - - - - - -   as shown in the 9x9 proof, therefore it takes
- - - - - - - - - -   a total of 16 knights to cover all patterns.
- - - - - - - - - -
- - - - - - - - d -
- - - - - - - - d d
c c c c - - - - d d
c c c - - - - - d d