[Retros] No (1+1) SPG at x=5

[Francois Labelle]

The month-long computer analysis of massacre proof games with 5
non-captures (x=5) that I was running is complete. I found no proof game
with only two kings on the board, with or without A1.3.

The best I get is one diagram with 2 solutions (with or without A1.3). I
checked the solutions and they're almost identical, so the problem is
cooked.


For PGs with 3 units left on the board I get:

- 100 problems without A1.3
- 13 of these are "disabled" by A1.3
- 2 new problems are "enabled" by A1.3 (hi Andrew!)
- for a total of 89 problems with A1.3
(Recall that for x=4 there were 18 PGs without A1.3, and 17 PGs with
A1.3.)


I verified the two x=5 problems of the Chess Problem Database whose status
was unknown:

P0000284 by Karlheinz Bachmann is C+ by my program
P1000210 by Thierry Le Gleuher is C+ by my program (also C+ by Popeye
after 46 min)


Here are statistics about the interesting PGs I found along the way:

- 26170 at-home
-   135 stalemate
-     7 vertical symmetry with KQ-equivalent
-  1400 horizontal symmetry and color-flip
-   428 180-degree symmetry and color-flip


Some individual SPGs of possible interest:

(1+1) in 17.5  (despite being cooked with 2 solutions)

(1+2) in 17.0  SPG enabled by A1.3 (428 solutions w/o A1.3)
(2+1) in 17.0  "                   (6216 solutions w/o A1.3)

(7+7) in 11.5  at-home AND mirror symmetric
(7+7) in 11.5  "
(5+5) in 13.5  "
(4+4) in 14.5  "
(3+3) in 15.5  "

(3+1) in 16.5  at-home (most depleted)

(7+3) in 13.5  stalemate (least depleted)
(1+4) in 16.0  stalemate (most depleted)

(12+10) in 7.5  vertical symmetry with KQ-equivalent
(10+10) in 8.5  " (this one has also a variant)
(6+8)  in 11.5  "
(6+4)  in 13.5  "
(4+2)  in 15.0  " (this one is x=4)

(2+2) in 16.5  180-degree symmetry and color-flip (most depleted)


I didn't backtrack every solution yet. I plan to put the (1+1) problems on
my website and post the interesting SPGs and their solutions to the
mailing list in a few days, unless someone contacts me with a more
glamorous publishing option. Some of the positions are pretty stunning,
but I haven't found the "holy grail".

	Francois