Efficient Knight Cover Patterns Part I
The Knight Over-Coverage Puzzle
Introduction
This essay describes a technique that has been successful in finding efficient knight covering patterns.
These patterns all show a common theme, leading to a method of directly generating them all.
Knight Coverage and Over-Coverage
A single knight can cover 9 squares of the board,
but in optimal solutions on the smaller boards, you need about one knight for every 6 squares.
Obviously, knight coverage is being wasted.
If a knight is placed near the side of the board, some of the knight coverage spills over the edge.
But not many knights do this in optimal solutions.
The wasted coverage is due mainly by squares being covered more than once.
These squares are said to be over-covered.
For a solution on an NxN board dominated by D knights,
NxN = 9xD - over-coverage - spillage
or
D = (over-coverage + spillage - NxN) / 9.
Since NxN and 9 are constants for a particular board size,
minimizing the dominating knights, D, is equivalent to
minimizing the sum of over-coverage and spillage.
As the boards get larger, the efficiency in the corners and edges does not change very much.
The opportunity for greater efficiency is only in the center.
So spillage over the edge and over-coverage near the edge become virtually negligible.
Thus, as the boards get larger,
the Knight Cover Puzzle is asymptotically to minimize the over-coverage.
Finding Efficient Knight Cover Patterns
Looking for a pattern that covers 8 squares per knight is the same as
looking for a pattern with an over-coverage of 1 square per knight.
Enumerating optimal solutions for small to medium board sizes doesn't seem to get very far.
A lot of area in the center is needed before a high efficiency pattern can be effective.
The 18x18 board is the smallest size where efficient patterns become usable enough
to demonstrate their extendability.
Since not all of the patterns appear on the 18x18, we would need to check larger boards.
We need some patterns that will work on odd-sized boards, too.
But enumerating solutions for 18x18 boards and larger takes too much time.
Fortunately there is a better way.
There is a variation of the knight cover puzzle in which high efficiency patterns
appear in extendable form on much smaller board sizes.
In this new puzzle, 8x8 is the smallest board where an efficient pattern first appears.
Monier's pattern first appears on the 8x9 size.
Patterns not seen at all in smaller boards for the Knight Cover Puzzle
also appear on the 8x9 size of the new puzzle.
These board sizes are small enough so that enumerating their optimal solutions
is a feasible method of finding new patterns and, hopefully, all efficient patterns.
The Knight Over-Coverage Puzzle
Since the essential objective of the Knight Cover Puzzle is to minimize the over-coverage,
perhaps a puzzle that more directly addresses this goal will be better at producing the patterns we seek.
Imagine a 12x12 board containing an 8x8 board at its center.
It is desired to place knights to completely cover the 8x8 board.
Knights may be placed either on the 8x8 board or outside of it.
This requires an extra two rows and two columns on all sides of the 8x8 board,
hence the 12x12 board.
A square on the 8x8 board is covered if it is occupied or attacked.
It is over-covered by 1 if it is covered by two knights;
it is over-covered by 2 if it is covered by three knights, and so on.
How do you place knights to completely cover the 8x8 board while minimizing the total over-coverage?
There is no limit to the number of knights used.
Only squares on the 8x8 board count toward total over-coverage.
Here is an optimal solution for the 7x7 board.
| X | | | | | | X | |
| | X | | | | | | | |
X | - X - - - - - | |
| | - - - - - X X | X
| | - - - - - - - | |
| | - - - X - - - | |
| | - - - - - - - | |
X | X X - - - - - | |
| | - - - - - X - | X
| | | | | | | | X | |
| | X | | | | | | X |
The over-coverage for the 7x7 is 0.
That is, 7x7 and smaller boards do not require any over-coverage.
The 8x8 board is the smallest square board that requires some over-coverage.
The minimum is 4. There are two solutions.
The diagram to the right of a solution shows
the location of the over-covered squares and the amount of over-coverage.
8x8 solution 1 of 2
(not extendable)
| | | | X | | X | | | | | | | | | | | | | | | |
| | X | | | | | | X | | | | | | | | | | | | | |
| | - - - - - - - - | | | | - - - - - - - - | |
X | - - - - - - - - | X | | - - - - - - - - | |
X | - - - - - - - - | X | | - - 1 - - 1 - - | |
| | - - - X X - - - | | | | - - - - - - - - | |
| | - - - X X - - - | | | | - - - - - - - - | |
X | - - - - - - - - | X | | - - 1 - - 1 - - | |
X | - - - - - - - - | X | | - - - - - - - - | |
| | - - - - - - - - | | | | - - - - - - - - | |
| | X | | | | | | X | | | | | | | | | | | | | |
| | | | X | | X | | | | | | | | | | | | | | | |
8x8 solution 2 of 2
(showing part of the extendable opposing cups pattern)
| | | | | | | | | | | | | | | | | | | | | | | |
| | | | | C C | | | | | | | | | | | | | | | | |
X | - - - - - - - - | X | | - - - - - - - - | |
| | - - C - - C - - | | | | - - 1 - - 1 - - | |
| | - - - - - - - - | | | | - - - - - - - - | |
| | C - - - - - - C | | | | - - - - - - - - | |
| | C - - - - - - C | | | | - - - - - - - - | |
| | - - - - - - - - | | | | - - - - - - - - | |
| | - - C - - C - - | | | | - - 1 - - 1 - - | |
X | - - - - - - - - | X | | - - - - - - - - | |
| | | | | C C | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | |
The over-coverage is only 4 because the board is small enough so that knights from the outside can cover most of it.
As we go to larger board sizes, more knights within the board must fend for themselves
and must start to form an efficient cover pattern.
Since there are no corner and edge restrictions, it can use an efficient pattern right away.
In the Knight Cover Puzzle, solutions get more efficient as the boards get larger.
In the Knight Over-Coverage Puzzle, solutions get less efficient.
In both puzzles, most of its score comes from the over-coverage in the center.
Thus both puzzles asymptotically approach the same objective and, hopefully,
the same set of solutions, i.e., the efficient patterns we seek.
The Knight Cover Puzzle requires a very large board before it comes anywhere near the asymptotic limit.
But for the Knight Over-Coverage Puzzle, we can already get a glimpse of it on the 8x8 board.
For the 8x9 board, we get a lot more than a glimpse.
We get some new cup patterns, Monier's pattern, and a lattice pattern.
The over-coverage is 6, shown in the companion diagrams.
8x9 (part of the aligned cups pattern)
| | | | | | | | | | x | | | | | | | | | | | | | | |
C | | C | | | | C | | | | | | | | | | | | | | | | |
| | - - - - - - - - - | | | | - - - - - - - - - | |
| C C - - - - - - C C | | | | 1 - - - - - - 1 - | |
| | - - - - - - - - - | | | | - - - - - - - - - | |
| | - - C - - C - - - | x | | - - 1 - - 1 - - - | |
| | - - - - - - - - - | | | | - - - - - - - - - | |
| | - - - C C - - - - | | | | - - - 1 1 - - - - | |
| | - - - - - - - - - | | | | - - - - - - - - - | |
C | - C - - - - C - - C | | | - - - - - - - - - | |
| | | | | | | | | | | | x | | | | | | | | | | | | |
| C | | | | | | | | C | | | | | | | | | | | | | | |
8x9 (part of the opposing cups pattern)
| | | | | | | | | | | | | | | | | | | | | | | | | |
| | | | | | C C | | | | | | | | | | | | | | | | | |
| | - - - - - - - - - | x | | - - - - - - - - - | |
| | - - - C - - C - - | | | | - - - 1 - - 1 - - | |
| C - - - - - - - - - | | | | - - - - - - - - - | |
| | - C - - - - - - C | | | | - 1 - - - - - - - | |
| | - C - - - - - - C | | | | - 1 - - - - - - - | |
| C - - - - - - - - - | | | | - - - - - - - - - | |
| | - - - C - - C - - | | | | - - - 1 - - 1 - - | |
| | - - - - - - - - - | x | | - - - - - - - - - | |
| | | | | | C C | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | | |
8x9 (Monier's pattern)
| | | | | | | | | | | | | | | | | | | | | | | | | |
x | | | | | | | | M M | | | | | | | | | | | | | | |
x | - M - - - - - - - | | | | - - - - - - - - - | |
| | - M - - - - - - - M M | | - 1 - - - - - - - | |
| | - - - M - - - - - | | | | - - - 1 - - - - - | |
| | - - - M - - - - - | | | | - - - 1 - - - - - | |
| | - - - - - M - - - | | | | - - - - - 1 - - - | |
| | - - - - - M - - - | | | | - - - - - 1 - - - | |
| M - - - - - - - M - | | | | - - - - - - - 1 - | |
| M - - - - - - - M - | x | | - - - - - - - - - | |
| | | M | | | | | | | | x | | | | | | | | | | | | |
| | | M | | | | | | | | | | | | | | | | | | | | | |
8x9 (lattice pattern)
| | | | | L | | | | | | | | | | | | | | | | | | | |
| | | | | | | | | L | | | | | | | | | | | | | | | |
L | - L - - - - - - - L | | | - - - - - - - - - | |
| | - - - - - L - - - | | | | - - - - - 1 - - - | |
| L - - - - - - - L - | | | | - - - - - - - 1 - | |
| | - - - L - - - - - | | | | - - - 1 - - - - - | |
| | - - - - - L - - - | L | | - - - - - 1 - - - | |
| | - L - - - - - - - | | | | - 1 - - - - - - - | |
| | - - - L - - - - L | | | | - - - 1 - - - - - | |
| L - - - - - - - - - | | | | - - - - - - - - - | |
| | | L | | | | L | | | | | | | | | | | | | | | | |
| | | | | | | | | | L | | | | | | | | | | | | | | |
There is something in common with all these patterns.
Every unoccupied square is covered exactly once.
That is, over-coverage only occurs on occupied squares.
9x9 requires an over-coverage of 7 and uses the opposing cups pattern.
10x10 gives nothing new.
11x11 also uses the same three patterns, but in a variety of ways.
It also is large enough to demonstrate that the patterns are all extendable
while maintaining the efficiency of 8 squares per knight.
11x11 requires an over-coverage of 13.
aligned Monier alternating aligned Monier
| | | | M | | | | | | | M | | | | M | | | | | | | | | | | |
| | | | M | | | | | | | M | | | | | | M | | | | | | M M | |
| | - - - - M - - - - - - | M | | - - M - - - - - - - - | |
| | - - - - M - - - - - - | M | | - - - - M - - - - - - M M
| | - - - - - - M - - - - | | | | - - - - M - - - - - - | |
| M - - - - - - M - - - - | | M | - - - - - - M - - - - | |
| M - - - - - - - - M - - | | | | - - - - - - M - - - - | |
| | - M - - - - - - M - - | | | M M - - - - - - - M - - | |
| | - M - - - - - - - - M | | | | - - - - - - - - M - - | |
| | - - - M - - - - - - M | | | | - M M - - - - - - - M | |
| | - - - M - - - - - - - | | | | - - - - - - - - - - M | |
| | - - - - - M - - - - - | X | | - - - M M - - - - - - | |
| | - - - - - M - - - - - | X | | - - - - - - - - - - - | X
X X | | | | | | | M | | | | | | | | | | | | M M | | | | | X
| | | | | | | | | M | | | | | | X | | | | | | | | | | | | |
aligned cups alternating aligned cups
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| | | X X | | | | | | X X | | | | | | C | | C | | | | X | |
| | - - - - - - - - - - - | | | | - - - - - - - - - - - | |
| X - - - - C - - C - - - | X | | - - - C C - - - - - - X X
| | - - - - - - - - - - - | | | | - - - - - - - - - - - | |
X | - - - - - C C - - - - | | | | - C - - - - C - - C - | |
| | - - - - - - - - - - - | | | C - - - - - - - - - - - | |
| | C - - C - - - - C - - C | | C - - - - - - - C C - - | |
| | - - - - - - - - - - - | | | | - C - - - - - - - - - | |
| | - C C - - - - - - C C | | | | - - - - - C - - - - X | |
| | - - - - - - - - - - - | | | | - - - C - - - - - - - | |
| X - - - - C - - C - - - | X | | - - - C - - - - - - - X X
| | - - - - - - - - - - - | | | | - - - - - C - - - - - | |
X | | | | | | C C | | | | | | | | | X | | | | | | | X | | |
| | | | | | | | | | | | | | | | X | | | | | | | X | | | | |
cup water wheel opposing cups
| | | | | | | | | | | | X | | | | | | | | | | | | | | | | |
| | | X X | | | | | | | | | X | | | | C | | C | | | | X | X
| | - - - - - - - - X - - X | | | - - - - - - - - - - - | X
| X - - - - C - - - - - - | | | | - - - C C - - - - - - | |
| | - - - - - - C - - - - | | | | - - - - - - - - - - X | |
| | - - - - - - C - - - - | | | | - C - - - - C - - - - | |
| X - - - - C - - - - - - | | | C - - - - - - - - C - - | |
| | - - - - - - - - C - - C | | C - - - - - - - - C - - | |
| | - C C - - - - - - - - | | | | - C - - - - C - - - - | |
| | - - - - - - - - - C C | | | | - - - - - - - - - - X | |
| | C - - C - - - - - - - | | | | - - - C C - - - - - - | |
| | - - - - - - - C - - - | X | | - - - - - - - - - - - X X
X | - - - - - C - - - - - | | | | - - C - - C - - - - - | |
| | | | | | | C | | | | | | | | | | | | | | | | | | X | | |
| X | | | | | | | C | | | | | | X X | | | | | | X | | | | |
aligned lattice alternating aligned lattice
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| | | | X | | L | | | | X | | | | | | X | | L | | | | X | |
| | - - - - - - - - - - - | | | | - - - - - - - - - - - | |
| L - - - - L - - L - - - | X | X - - - - L - - L - - - | X
| | - - - - - - - - - - - | | | | - - - - - - - - - - - | |
L | - L - - - - L - - L - | | | | - L - - - - L - - L - | |
| | - - - - - - - - - - - | | | L - - - - - - - - - - - | |
| | L - - L - - - - L - - L | | | - - - L - - - - L - - L |
| | - - - - - - - - - - - | | | | - L - - - - - - - - - | |
| | - - L - - L - - - - L | | | | - - - - - L - - - - L | |
| | - - - - - - - - - - - | | | | - - - L - - - - - - - | |
| X - - - - L - - L - - - | X | X - - - - - - - L - - - | X
| | - - - - - - - - - - - | | | | - - - - - L - - - - - | |
| | | X | | | | L | | X | | | | | | X | | | | | | | L | | |
| X | | | | | | | | | | | | | | X | | | | | | | L | | | | |
Is 8 Squares Per Knight The Most Efficient Pattern?
The two puzzles converge to the same solution for large boards, one from below, the other from above.
8 squares per knight is the lower bound, based on actual solutions from the Knight Cover Puzzle.
The Knight Over-Coverage Puzzle appears to converge to the very same patterns.
If there were a better than 8 squares per knight pattern, the Knight Over-Coverage Puzzle would always be using it.
But this does not seem to be the case.
However, the board sizes shown above are very small and much of the efficiency
comes from knights attacking from outside the board.
There is a possibility that a larger board size or a rectangular board with the right dimensions
will have a pattern that is slightly better than 8 squares per knight,
but does not show up in an optimal solution of smaller boards
because it doesn't have an efficient match with the board dimensions.
If a more efficient pattern exists,
it will eventually take over all the optimal solutions on large enough board sizes.
The size of the board it would take before this happens
depends on the size of its tile and how much more efficient it is.
The 7x7 solution offers a clue to the possibility of another efficient pattern.
It consists of a single knight that alone covers 9 squares.
It is surrounded by other knights which cover all the other squares without over-covering any of its 9 squares.
If this pattern can be extended and repeated to another such knight,
then slightly better than 8 squares per knight can be achieved.
Optimal solutions on 9x9 and 11x11 boards show sections of the 7x7 pattern,
but those boards are still too small to be sure that this pattern is efficient and extendable.
Somebody needs to enumerate solutions for larger boards to see if this pattern survives.
Generating All Efficient Cover Patterns
After extending the patterns and examining them,
we confirm that the theme we observed earlier has been maintained.
The over-covered squares are only the occupied ones.
In other words, every square is covered by an attack.
If every square is attacked in a solution,
then this solution also works for the Attack-Only Knight Cover Puzzle,
which requires squares to be covered only by attacking them;
occupying a square doesn't count as a cover.
Since knights attack squares only of opposite color,
the Attack-Only Knight Cover Puzzle is really two completely separate puzzles;
one puzzle to cover the white squares, the other puzzle to cover the black squares.
On a finite board with an even number of squares,
the white square puzzle shape is identical to the black square puzzle shape,
thus there is really only one puzzle, half the size.
Once we solve the white square puzzle, we can also use the solution for the black squares.
But the black square solution can be rotated and reflected in up to 8 different ways
before combining it on the original board.
So each individual solution has up to 8 different looking patterns when seen on the full board.
On an infinite board, or for a tilable solution pattern,
we can also translate the black square solution before combining it,
creating even more patterns for the full board.
Consider the following two white square solutions.
They both consist of a tilable pattern of strands on every other square of a diagonal, each strand 8 squares away.
The only difference between the two solutions is that a strand may alternate which rows they are on.
W - - - - - - - W - - - - - - - - W - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - -
- - W - - - - - - - W - - - - - - - - W - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - -
- - - - W - - - - - - - W - - - - - - - - W - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - -
- - - - - - W - - - - - - - W - - - - - - - - W - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W -
W - - - - - - - W - - - - - - - W - - - - - - - - W - - - - - - - -
- - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - - - -
- - W - - - - - - - W - - - - - - - - - - - - - - - - W - - - - - -
- - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - -
- - - - W - - - - - - - W - - - - - - - - - - - - - - - - W - - - -
- - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - -
- - - - - - W - - - - - - - W - - - - - - - - - - - - - - - - W - -
- - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - -
W - - - - - - - W - - - - - - - W W - - - - - - - - - - - - - - - W
The knights on these white squares attack every black square exactly once.
Thus they form a complete and efficient cover solution.
We can form the very same two solutions for the black knights and combine them on the same board.
Since the black knight solution is completely independent of the white knight solution,
we can put the black knight solution any where we want and in any orientation.
Also, since there are two solution variations for black, this results in four combinations.
If we keep the same orientation and put black knights in contact with the white knights,
we get two variations of Monier's Pattern.
W B - - - - - - W B - - - - - - - W B - - - - - - - B - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - -
- - W B - - - - - - W B - - - - - - - W B - - - - - - - B - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - -
- - - - W B - - - - - - W B - - - - - - - W B - - - - - - - B - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - -
- - - - - - W B - - - - - - W B - - - - - - - W B - - - - - - - B -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W -
W B - - - - - - W B - - - - - - W - B - - - - - - W B - - - - - - -
- - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - - - -
- - W B - - - - - - W B - - - - - - - - B - - - - - - W B - - - - -
- - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - -
- - - - W B - - - - - - W B - - - - - - - - B - - - - - - W B - - -
- - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - -
- - - - - - W B - - - - - - W B - - - - - - - - B - - - - - - W B -
- - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - -
W - - - - - - - W B - - - - - - W W - - - - - - - - B - - - - - - W
If we translate the black knights four squares to the right,
we get two variations of the lattice pattern.
W - - - - B - - W - - - - B - - - W - - - - B - - - - - - - B - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - -
- - W - - - - B - - W - - - - B - - - W - - - - B - - - - - - - B -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - -
- B - - W - - - - B - - W - - - - - B - - W - - - - B - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - -
- - - B - - W - - - - B - - W - - - - - B - - W - - - - B - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W -
W - - - - B - - W - - - - B - - W - - - - - B - - W - - - - B - - -
- - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - - - -
- - W - - - - B - - W - - - - B - - - - - - - - B - - W - - - - B -
- - - - - - - - - - - - - - - - - B - - W - - - - - - - - - - - - -
- B - - W - - - - B - - W - - - - - - - - - - - - - B - - W - - - -
- - - - - - - - - - - - - - - - - - - B - - W - - - - - - - - - - -
- - - B - - W - - - - B - - W - - - - - - - - - - - - - B - - W - -
- - - - - - - - - - - - - - - - - - - - - B - - W - - - - - - - - -
W - - - - B - - W - - - - B - - W W - - - - - - - - - - - - B - - W
If we make a 1/4 turn of the black solution so that its diagonal strands run perpendicular to the white strands,
we get the aligned cups pattern and the alternate aligned cups.
W B - - - - - - W B - - - - - - - W B - - - - - - - B - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - -
- - W - - - - B - - W - - - - B - - - W - - - - B - - - - - - - B -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - - - -
- - - - W B - - - - - - W B - - - - - - - W B - - - - - - - B - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W - - -
- - - B - - W - - - - B - - W - - - - - B - - W - - - - B - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W -
W B - - - - - - W B - - - - - - W - B - - - - - - W B - - - - - - -
- - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - - - -
- - W - - - - B - - W - - - - B - - - - - - - - B - - W - - - - B -
- - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - - - -
- - - - W B - - - - - - W B - - - - - - - - B - - - - - - W B - - -
- - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - - - -
- - - B - - W - - - - B - - W - - - - - B - - - - - - - B - - W - -
- - - - - - - - - - - - - - - - - - - - - - - - W - - - - - - - - -
W B - - - - - - W B - - - - - - W W B - - - - - - - B - - - - - - W
Using the second solution for both white and black knights, so that strands from both solutions alternate rows,
we get the opposing cups/cup water wheel pattern.
W - - - - - - - - B - - - - - - -
B - - - - - - - - W - - - - - - B
- - W - - - - B - - - - - - - - -
- - - - - - - - - - - W - - B - -
- - - - W B - - - - - - - - - - -
- - - - - - - - - - - - B W - - -
- - - B - - W - - - - - - - - - -
- - - - - - - - - - B - - - - W -
- B - - - - - - W - - - - - - - -
- W - - - - - - B - - - - - - - -
- - - - - - - - - - W - - - - B -
- - - W - - B - - - - - - - - - -
- - - - - - - - - - - - W B - - -
- - - - B W - - - - - - - - - - -
- - - - - - - - - - - B - - W - -
- - B - - - - W - - - - - - - - -
W - - - - - - - - B - - - - - - W
Note that the second solution that alternates rows for the strands is really a family of solutions.
For each strand, any of two row placements may be chosen.
Thus there are actually an infinite variety of strand sequences.
When various combinations of strand sequences are chosen for the white and black knight solutions,
and then combined perpendicularly, they result in an infinite variety of cup patterns.
At one time, people thought that Monier's Pattern was the only one that achieved 8 squares per knight.
In a certain sense, they were right because all the patterns are isometries of Monier's Pattern.
But in a practical sense, when these patterns are used to fit a particular board size,
no two are the same.