[Retros] Proof games with loss of tempo

[Francois Labelle]

For each value of t, what is the smallest n with a PG that has
a) a unique solution in n moves and some solution(s) in n - t moves?
b) a unique solution in both n and n - t moves?

Version (b) is harder and more "publishable", but (a) is also
interesting from a mathematical point of view. Below are some results.

The first two values of t are easy:

t=0.5
a,b) n=1.5: 8*20 = 160 combinations of a white pawn double-jump and any
black move.

t=1.0
a,b) n=2.0: 8*8 = 64 combinations of a white pawn double-jump and a
black pawn double-jump.

Here are the results of a computer search up to 5.0 moves:

t=1.5
a) n=4.0: 1.g3 d6 2.g4 Bxg4 3.h3 Bf3 4.Nxf3 d5
b) n=4.5: 1.d3 g5 2.Bxg5 h6 3.Bxh6 Rh7 4.Bg7 Rxg7 5.d4
(b) has the additional property that the solution is unique in 3.0, 3.5,
4.0, and 4.5 moves, so that question is settled too. (Andrew Buchanan
claimed to have a solution to that problem on
http://anselan.com/STAsols.html but it might not have been shortest.)

t=2.0 (impossible)

t=2.5
a,b) n=5.0: 1.d4 c5 2.Bf4 c4 3.Bc7 c3 4.Nxc3 Nf6 5.Nb1 Nd5
That problem was used in the Messigny 2011 retro solving competition
http://www.pairlist.net/pipermail/retros/2011-June/003569.html .

And here are current records that I could find on the Chess Problem
Database:

t=3.0
a,b) n=10.0 (Michel Caillaud, 3074 Probleemblad 09/1996, 1st prize)
http://www.softdecc.com/pdb/search.pdb?expression=PROBID=%27P0008478%27

t=3.5 ?

t=4.0 (impossible)

t=4.5
a,b) n=12.0 (Gerd Wilts, R226 Probleemblad 02/2004)
http://www.softdecc.com/pdb/search.pdb?expression=PROBID=%27P1013079%27

Any other result?

	Francois