Theorem An infinite board knight cover with 9 squares per knight is not possible. Proof This can only happen if knights cover disjoint sets of 9 squares. That is, there can't be any squares that are over-covered. To show that this is not possible, here is an attempt to cover a board without an over-covered square. Place the first knight on the W-square and consider covering the o-squares of Figure 1. We can't occupy two of the o-squares because the two knights will have attacked squares in common. So at least one o-square must be covered by attack. The x-squares are the only locations that can attack an o-square while avoiding squares attacked by the W knight. - - - - - - - - - - - - - - x - - - - - - B - - - - - B x - - - x - - - - W - - - - W x x - - - - W B o - - - W - - o - - - - - - - - - - o - - - - o - - - - - - - - - - o W o - - o W - - - - W - - - - - - W - - - - - W - - - - o - - - - o - - - - - - - - - - - - - - - - - - - - x - - - x x - - - - W - - - - - - W - - - - - W - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Occupying an x-square results in Figure 2, which also shows the remaining two o-squares that still need to be covered. As before, the o-squares can't both be occupied, so at least one needs to attacked. The x-square is the only location available that can attack without creating an over-covered square. The rest of the figures show the result of occupying an x-square from the previous figure using W for a white-squared knight or B for a black-squared knight. The x-squares are the only locations that can cover an o-square without creating an over-covered square. This process ends in Figure 5, which shows an o-square that can't be covered from anywhere without creating an over-covered square.