This page contains a solution for the attack-only knight covering of an infinite board.
In this puzzle, all squares must be attacked.
Occupation of a square does not count as a cover.
The relation of this to the regular knight cover puzzle is that on the infinite board,
there is a strong conjecture that both puzzles have the same set of optimal solutions.
Each knight can attack at most 8 squares.
If every knight of a complete solution attacked a unique set of 8 squares,
then that would be an optimal solution.
It would also mean that no square is attacked more than once.
The following will show that this is indeed possible.
There must be knights on the board and w.l.o.g.
we can assume there is at least one white-squared knight.
Figure 2 shows a white-squared knight indicated by the W-square.
The two o-squares need to be attacked by other knights.
- - - - - - - - - - - - - -
- - - - - - - - - y - y - -
- - - - - - - - x - y - x -
- - - o - - - y - - o y - -
- - o W - - - - y o W - y -
- - - - - - - y - y - y - -
- - - - - - - - x - y - - -
- - - - - - - - - - - - - -
Figure 2 Figure 3
Figure 3 shows all the locations where a white-squared knight can attack an o-square.
These are indicated by the x-squares and y-squares.
But note that if a knight occupies a y-square,
then its attacks will combine with the W-square knight
and squares will be attacked more than once.
This will not happen when occupying an x-square.
So one of the x-squares must be occupied.
Each x-square is 2 diagonal squares away from the W-square.
So no matter which x-square you pick, you can rotate the board
so that the two white-squared knights will look like Figure 4.
Figure 4 also shows another two o-squares that need to be attacked.
The x-square is the only location where a knight can attack an o-square
and not leave another square attacked more than once.
- - - - - - - -
- - - - - - - -
- - - - - x - -
- - - o - - - -
- - - W o - - -
- - - - - - - -
- W - - - - - -
- - - - - - - -
Figure 4
Figure 5 shows the resulting pattern, a strand of knights
placed on every other square along a diagonal.
The logic of Figure 4 can be repeatedly applied to the ends of the strand,
extending it in both directions forever.
- - - - - - - - x
- - - - - - o - -
- - - - - - W o -
- - - - - - - - -
- - - - W - - - -
- - - - - - - - -
- o W - - - - - -
- - o - - - - - -
x - - - - - - - -
Figure 5
Figure 6 shows a segment of an infinitely long diagonal strand.
It also shows two o-squares which need to be attacked.
However, the only locations for another knight that won't over-attack a square
are the three x-squares.
So one of the x-squares must be occupied.
- - - - - - - - - - - -
- - - - - - - W - - - -
- - - - - - - - - - - -
- - - - - W - - - - - -
- - - - - - - - - - - -
- - - W - - - - o - - -
- - - - - - - o - - x -
- W - - - - - - - x - -
- - - - - - - - x - - -
- - - - - - - - - - - -
Figure 6
There are two essentially different choices of an x-square
as shown in Figures 7A and 7B.
In each figure, another pair of o-squares needs to be attacked
and can only be done by the shown x-squares without over-attacking a square.
- - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - W - - - - - - - - - - - - W - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - W - - - - - - - - - - - - W - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - -
- - - W - - - - - - - x - - - - W - - - - - - - - -
- - - - - - - - - o - - - - - - - - - - - - - x - -
- W - - - - - - o W - - - - W - - - - - - o - - - -
- - - - - - - - - - - - - - - - - - - - o W - - - -
- - - - - - - x - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - x - - - - - -
Figure 7A Figure 7B
Choosing either x-square leaves the start of another diagonal strand.
As before, this must be extended to infinity in both directions along the diagonal,
resulting in Figures 8A and 8B.
Both diagrams show two diagonal strands, 8 squares apart,
but Figure 8A shows them on the same rows and
Figure 8B shows them on different rows.
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - W - - - - - - - - - - - - W - - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - W - - - - - - - - - - - - W - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - W - - - - - - - W - - - - W - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - W -
- W - - - - - - - W - - - - W - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - W - - -
- - - - - - - W - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - W - - - - -
Figure 8A Figure 8B
Applying the logic from Figure 6, we must have an infinity of strands, 8 squares apart,
but the knights comprising each strand can be either on the even rows or the odd rows.
Figures 9A and 9B show all the squares that are attacked by the strand pattern.
Note that all the black squares are attacked.
This means that there is no other square for an additional white-squared knight
because it will over-attack a black square.
But not all squares of the board are attacked.
so we must add some black-squared knights.
- a - a - a - a - a - a - - a - a - a - a - a - a
a - a - a - a W a - a - a a - a - a - a W a - a -
- a - a - a - a - a - a - - a - a - a - a - a - a
a - a - a W a - a - a - a a - a - a W a - a - a -
- a - a - a - a - a - a - - a - a - a - a - a - a
a - a W a - a - a - a W a a - a W a - a - a - a -
- a - a - a - a - a - a - - a - a - a - a - a W a
a W a - a - a - a W a - a a W a - a - a - a - a -
- a - a - a - a - a - a - - a - a - a - a W a - a
a - a - a - a W a - a - a a - a - a - a - a - a -
- a - a - a - a - a - a - - a - a - a W a - a - a
Figure 9A Figure 9B
The same logic can be applied to the black-squared knights,
showing that they must also form the diagonal strand pattern.
The black-squared knight pattern can be combined with the white-squared knight pattern
in any placement or orientation.
This is true because white-squared knights only attack black squares and
black-squared knights only attack white squares,
so no square will be attacked more than once.
Black strands can be combined with the white strands either in parallel or perpendicular.
When in parallel and white and black knights are adjacent,
you obtain Monier's Pattern.
When in parallel and not adjacent, you obtain a sort of lattice pattern.
When they are perpendicular, they form a pattern of cups.
[Use the W/B diagrams from my essay here]
By selecting alternate rows for some of the strands,
a variety of pattern modifications can be produced.
In particular, the cups can be made to change their
orientation in an infinite variety of ways.