More on Displacement and Distance

Responding to my post on 'Bizarre Castling Rules'?, HarryO suggested examining the R***K**R positions (see Non-Random Chess960 Trial Game 9: SP864 after 4)...e6), e.g.

1) Do these 18 starts allow [full] Queenside and Kingside castling strategies or are they restricted in some way? [...]

with the further restriction that

The main culprits we should investigate are the six SPs with Knights on the same color probably starting with the four N*N SPs.

I had some difficulties to compare the different positions, so I created a chart in the same style I used in The KID Family Goes the Distance.

With the left Rook (zLR), King, and right Rook (zRR) fixed to their original squares, the list includes SP518, while the 'distance' (Dist) from that position to the other positions is typically low. It turns out that HarryO's suggested positions can be identified easily from a calculation based on the displacements of the pieces (zLR, zLN, etc.).

For example, the first position in the list, SP414, has the left Knight (zLN) displaced by one square and the right Knight (zRN) by three squares. The sum of these -- four squares -- is an even number; the Knights start on different color squares, because they start on different color squares in SP518. For the second position in the list, SP430, zLN = 0 and zRN = -3. The sum of these displacements (using their absolute values) is an odd number, so the Knights are on the same color square (N*N in SP430).

[By similar reasoning, the combined displacements of the Bishops must always be an even number. The Bishops are on different color squares in SP518, a requirement that holds for all chess960 positions.]

Of the 18 positions in the R***K**R family, HarryO is right in pointing out that only six have the Knights on the same color. That means twelve positions have Knights on different colors. Why the imbalance in the two cases? Because Knights on the same color mean there are fewer possibilities for placing the Bishops.

Another curiosity of the same type is the number of positions in the family where a Knight starts on the same square as SP518. There are eight positions where zLN = 0, but only six where zRN = 0. Why the difference? Because in R***K**R, the left Rook and the King are on the same color as the right Knight. This again limits the choice for placing one of the Bishops.

That's all nice to know, but it doesn't tell us anything about the dynamics of the different SPs. Some practical tests will be necessary to do that.