PolyKnights

 

To quote from Wikipedia:

A polyknight is a plane geometric figure formed by selecting cells in a square lattice that could represent the path of a chess knight in which doubling back is allowed. It is a polyform with square cells which are not necessarily connected, comparable to the polyking. Alternatively, it can be interpreted as a connected subset of the vertices of a knight's graph, a graph formed by connecting pairs of lattice squares that are a knight's move apart.

It would seem that the definition should read "doubling back is not allowed" since none of the examples shown features "doubling back" in the sense of returning to a previously occupied square.

Today I turned 26379 days and this number happens to be a member of OEIS A030446:

A030446

Number of \(n\)-celled polyknights (polyominoes connected by knight's moves).

In the comments to this sequence, it is stated that:

A polyknight is a variant of a polyomino in which two tiles a knight's move apart are considered adjacent. A polyknight need not be connected in the sense of a polyomino. These are free polyknights.

By free here is meant that the pieces can be picked up and turned over, as opposed to one-sided pieces that cannot be picked up and flipped over or fixed pieces that cannot be moved at all. Figures 1 and 2 help explain the differences, using tri-knights are examples:

Figure 1

Figure 2

The six possible tri-knights are shown in Figure 3:

Figure 3

Figure 4 shows a table from Wikipedia listing the different numbers of free, one-sided and fixed polyknights:

Figure 4

As can be seen, the sequence for \(n\) polyknights runs: 1, 1, 6, 35, 290, 2680, 26379 and the numbers very large very quickly. However, for one-sided and fixed polyknights, they rise ever more quickly. The shapes when \(n=7\) could be called heptaknights. Figure 5 shows three examples out of the 26379 possibilities for free heptaknights.

Figure 5

It's also apparent from the sequence terms in Figure 4 that I won't encounter any more polyknight days in this lifetime. The next free polyknight number is 267,598, the next one-sided polyknight number is 52,484 and the next fixed polyknight number is 209,608. 

For information on polykings, follow this link: https://mathworld.wolfram.com/Polyplet.html