# [Retros] Happy New Year!

Francois Labelle flab at EECS.Berkeley.EDU
Tue Jan 13 02:36:32 EST 2004

Noam Elkies is right, I can search for positions with n solutions for up
to ply 8.

For n=2004, there is a proof game in 7 plies:

r n b q k b n r
p p p p . p p p
. . . . p . . .
. . . . . . . .
. . . . . . . .
. . P . P . . .
P P . P . P P P
R N B Q K B N R

(with the requirement that castling is still possible for both sides)

Unfortunately, with examples found by computer there is no guarantee that
counting the number of solutions is easy for a human, but in this case
we're probably lucky that it's not painfully hard.

Here's my reasoning: the position can obviously be obtained in 3 plies, so
White and Black must each waste 2 plies by moving an officer forward and
backward. For each side the officer can be a knight, the f-bishop, or the
queen, but not the king because of the requirement that castling is still
possible.

We can count White's moves and Black's moves independently, and then take
into account contention for the squares a3 and a6.

Skipping the details, I get 92*22 - 10*1 - 2*3 - 2*2 = 2004.

In fact there are 4 such proof games for ply 7: the problem also works if
one or both e-pawns did a double jump.

There are 2232 proof games with 2004 solutions for ply 8.

Francois Labelle