More on the Concept of Distance

In Randomness in Chess960 Start Positions (SPs), I noted that starting from the traditional start position (SP518: RNBQKBNR) there are exactly three other SPs that are a straightforward swap of two adjacent pieces, and I assigned them a distance of one (see that previous post for an explanation of the distance calculation). The next level of complexity, distance two from SP518, arises from slightly more complicated piece swaps.

• The first scenario is where two pieces separated by a single square are swapped. An example is SP518 with the Queenside Rook and Bishop swapped (SP226: BNRQKBNR).
  
  
• A second scenario is where two pairs of adjacent pieces are swapped. An obvious example is swapping both Rooks with the neighboring Knight (SP326: NRBQKBRN).
  
  
• A third scenario is where three adjacent pieces rotate one square within their little group. An example is shifting the Rook and Knight one square to the right, then the Bishop taking the place of the Rook (SP514: BRNQKBNR).

I imagine that the distance of a particular SP from SP518 has something to do with the number of moves to be played before the game starts to look like a 'normal' game starting from SP518. The maximum distance is ten, and there are nine such positions.

SP080: BBNNRKRQ
SP094: NNRKRBBQ
SP095: NNRKRQBB
SP879: QRKRNNBB
SP912: BBRKRQNN
SP928: BBRKRNQN
SP944: BBRKRNNQ
SP958: RKRNNBBQ
SP959: RKRNNQBB

It's worth noting that all nine SPs have the three castling pieces in the tightest possible configuration ('RKR'), and all but one (SP928) have the Bishops and Knights starting next to the piece of the same type. This gives them less of a random look than when the minor pieces are scattered haphazardly around the back rank.

I compared the nine SPs against my database of games played at Mainz, hoping to find an example game between two top masters. The only match I found was the following 2006 simultaneous game between supergrandmaster Levon Aronian and supermodel Carmen Kass.

Mainz 2006, Chess960 Simul
Kass, Carmen

Aronian, Levon
(After 11.b2xc3(xN))
[FEN "rkr2qbb/ppp1p2p/2n3p1/4Pp2/3N1P2/2P3P1/P1P4P/RKR2QBB b - - 0 11"]

The diagrammed position is obviously not from a traditional SP518 opening. With a Bishop buried on h8 and a vulnerable King, Black appears to be in trouble. Instead of 11...Nxd4, making the best of a dubious position, Black blundered with 11...Na5, losing material. Here is a partial game score, courtesy of Chesstigers.de.

[Event "CCM6 - Chess960 Simultan"]
[Site "Mainz"]
[Date "2006.08.16"]
[Round "?"]
[White "Aronian, Levon"]
[Black "Kass, Carmen"]
[Result "1-0"]
[SetUp "1"]
[FEN "rkrnnqbb/pppppppp/8/8/8/8/PPPPPPPP/RKRNNQBB w CAca - 0 0"]

{SP959} 1.f4 f5 2.g3 g6 3.Nc3 Nd6 4.e4 Nc6 5.e5 Nc4 6.Nf3 d6 7.d3 Nb6 8.d4 dxe5 9.dxe5 Nd5 10.Nd4 Nxc3+ 11.bxc3 Na5 12.Qb5 c6 13.Qxa5 Qd8 14. Qb4 Bd5 15.Bxd5 cxd5 16.Ne6 Qd7 17.Nc5 Qc6 18.a4 {Rest der Partie nicht nachvollziehbar} 1-0

For more about the event, see Chess Classic: Chess960 results, Anand and Aronian simuls. A photo of Kass just before the start of the game is in CCM6 Bulletin 04 [PDF],