# [Retros] 2011

Noam Elkies elkies at math.harvard.edu
Wed Jan 5 20:48:05 EST 2011

Olli Heimo writes:

> I had just a correspondance with Gianni Donati.

> He suggests that because 2011 is a prime number

> there can't be any proof game with 2011 solutions.

> I disagree with him. As 2003 was also a prime, but

> you did it anyway.

Thanks for this reminder (third year in a row!). There must exist
proof games with 2011 moves: the number is small enough that
F. Labelle's exhaustive database must contain some rather short
proof games that happen to have exactly that many solutions.
But that probably won't be a good problem for humans to solve
and appreciate. Here Donati is right that a prime target --
or more generally one with a large prime factor -- makes the task
harder; but Olli is also right that the difficulty is not
insurmountable. I fall back on the same mechanism I used in 2003;
2011 is farther away from the key number, so it took some more effort:

White Rc3 Sc2 Bb4 Qa4 Kf5 Bf1 Sg1 Rh1 Pa2 Pb3 Pc4 Pd2 Pe2 Pf3 Pg2 Ph2
Black Re6 Sa6 Bb5 Qh6 Kb8 Bf8 Sg8 Rh8 Pa7 Pb6 Pc5 Pd6 Pe7 Pf7 Pg7 Ph7

(shortest) proof game in 13.5: how many solutions?

NDE 4.1.11

+---a---b---c---d---e---f---g---h---+

| |

8 . -K . . . -B -S -R 8

| |

7 -P . . . -P -P -P -P 7

| |

6 -S -P . -P -R . . -Q 6

| |

5 . -B -P . . K . . 5

| |

4 Q B P . . . . . 4

| |

3 . P R . . P . . 3

| |

2 P . S P P . P P 2

| |

1 . . . . . B S R 1

| |

+---a---b---c---d---e---f---g---h---+
dia13.5 16 + 16

No captures and no lost time; this took Popeye 3.41 about 20 seconds
to solve and list all solutions.

I had hoped to extend this by one play with a customary checkmating
conclusion, but this seems difficult; e.g. replacing Black's last move
by ...Re4 and finishing with 14...Qf4# or 14...Qg6# failes because
the Queen could have come from g3. If the last move of White's
Q-side sequence were f4 then 14...Qh5# would work, but I was not
able to engineer that. I tried instead this setup:

White Rb4 Sb1 Bb2 Qa5 Kf5 Bf1 Sg1 Rh1 Pa4 Pb3 Pc3 Pd2 Pe2 Pf3 Pg2 Ph2
Black Re3 Sa6 Bb5 Qh6 Kb8 Bf8 Sg8 Rh8 Pa7 Pb6 Pc5 Pd7 Pe7 Pf7 Pg7 Ph7

(shortest) proof game in 13.5: how many solutions?

NDE 5.1.11

+---a---b---c---d---e---f---g---h---+

| |

8 . -K . . . -B -S -R 8

| |

7 -P . . -P -P -P -P -P 7

| |

6 -S -P . . . . . -Q 6

| |

5 Q -B -P . . K . . 5

| |

4 P R . . . . . . 4

| |

3 . P P . -R P . . 3

| |

2 . B . P P . P P 2

| |

1 . S . . . B S R 1

| |

+---a---b---c---d---e---f---g---h---+
dia13.5 16 + 16

hoping to append 14...e5 15 Rg4 Se7#. This is an isomorphic puzzle,
but for some reason Popeye had much more trouble with it, taking over
four hours! Perhaps it didn't realize the Rook paths were minimal.
So I figure the extended version might tie it up for days. Can a
specialized proof game solver do it more efficiently?

NDE