## 29 September 2012

### The KQR Corner Family

In Non-Random Chess 960 Trial Game 5: SP468, HarryO identified an important family of start positions (SPs).
This SP is part of 108 SPs that feature the combination of a Rook, a King and a Queen in the corner, leaving most of the opposite side of the board to the minor pieces and a lone rook.

His math works, too. If we reserve f/g/h for the KQR (in whatever order), that leaves five squares for the other pieces. After randomly placing one Bishop on the three remaining squares of the same color, then randomly placing the other Bishop on the two squares of the same color, that leaves three squares to place the remaining Rook. Placing the Knights on the two remaining squares doesn't change the number of permutations, so we have 3 x 2 x 3 = 18 different permutations of the five pieces for each combination of KQR. Since there are three of those (see next paragraph), there are exactly 54 positions with KQR in one corner. Doubling that to account for the other corner gives 108 SPs, which is HarryO's number.

It's also worth noting that in the 54 KQR-corner SPs, the Queen determines the position of the other two pieces. If the Queen is on the f-file, the pieces must be placed 'QKR'; Queen on the g-file forces 'KQR'; and Queen on the h-file leaves 'KRQ'.

The family of SPs is important because the KQR at such close proximity interfere with each other in their initial movements. They also have different objectives in the opening: the King seeks safety, the Queen seeks a safe development square, and the Rook wants a connection with the other Rook. On top of that, the three heavy pieces bunched together present a convenient target for the enemy Bishops, which are most likely to be on the other side of the board, ready to take up attacking positions after a move or two.

That SP468 post is in fact a game where HarryO and I have been struggling with one such position (SP468 RBBNNKRQ). I'll discuss it in more detail when we are through with our investigations.