Game-determining moves

R498 Alex Fishbein
(USA)
Find an orthodox game that ends with 7...Kxb7# (C+)

R499 François Labelle
(Canada)
Find an orthodox game that ends with 8.Qaxc2# (C+)

Preface

This is an HTML version of an article originally published in The Problemist Vol 25 No 8, March 2016, with some hyperlinks added. The article refers to two problems published in the Retros column reproduced here on the right.

Sometimes a single move can be a chess problem. One simple example is the problem “3...Qd4#, game?” and the solution: 1.f3 e5 2.Kf2 Qh4+ 3.Ke3 Qd4#. This problem type is part of the slightly broader category of synthetic games where the solver is asked to construct a game of a specific length that obeys specific constraints. Synthetic games are a variant of the more familiar proof games where the solver is asked to construct a game of a specific length that ends with a specific diagram. Synthetic games aren't as common as proof games because the lack of a diagram is very limiting. I even consider most synthetic game problems to be discoveries rather than compositions.

In this article I consider only checkmate moves written in algebraic notation that uniquely determine the game with no additional conditions. These problems can be further split into two categories depending on whether the last move uses disambiguation or not. (Disambiguation means a specification of the file and/or rank of departure of the moving piece, for example the “4” in the move “S4c6” when a second knight can move to c6.) The table overleaf lists all known such problems including new discoveries R498 and R499 from this month's Retros column, and five originals using disambiguation for this article.

Algebraic notation

The problems that I consider in this article depend on algebraic notation, so it's important to review the concept. FIDE describes algebraic notation in Appendix C of the Laws of Chess. The appendix's main purpose is to dictate how players should record their moves on their scoresheets and it allows for some sloppiness. For example, marking checks, checkmates, or even captures is optional. Because of this, four non-capturing problems from the table (3...Qd4#, 5...Rh1#, 7...Q1c1#, and 8.Qfa6#) are technically cooked under FIDE by solutions ending in captures that the solver chose not to notate.

An alternative is Standard Algebraic Notation (SAN) as defined in the Portable Game Notation (PGN) standard, which mandates marking checks, checkmates, and captures (among other things). Given that these markings are almost universally used in chess publications and chess software, I think that the PGN standard is a better formalisation of the notation that we see in practice. This standard is also very elegant because it specifies exactly one way to write a move and it handles every possible case, including the case where three or more identical pieces can move to the same square, which FIDE doesn't cover. For these reasons, I choose to interpret synthetic game problems under the PGN standard for algebraic notation.

Problems without disambiguation

In this category, the length record stood for 21 years until it was beaten by Stuart Rachels in 2015, and beaten again by Alex Fishbein with R498. It turns out that Americans Stuart, an IM and Alex, a GM are part of a group of very strong chess players led by Stuart who set out to find a (not necessarily unique) shortest game for each of the 840 possible mating moves without disambiguation. (A mate by Black such as 2...Qh4# is considered shorter than the mirror move 3.Qh5# by White.) Their efforts led to the discovery of two new game-determining moves, and also helped quantify how much of the search space has been explored. I already knew from an exhaustive computer search up to 6.0 moves that exactly 661 mating moves can be achieved in 6.0 moves (mate on Black's sixth) or fewer. Comparing notes, Stuart's group found the correct minimal length in 94% of the 661 cases.

Stuart's group found 74 more mating moves that they could achieve in 6.5 moves, which must also be the minimal length since they weren't found in my computer search. I also proved with a special program that the solution of R498 in 7.0 moves is unique and of minimal length. This gives a total of 661 + 74 + 1 = 736 mating moves (88%) for which we have a guaranteed shortest solution. If a mating move with a unique solution remains to be discovered, then it is almost certainly a move in the remaining 12% that admits a solution shorter than the one found by Stuart's group. Its existence seems unlikely given the small search space and the low error rate of Stuart's group.

Problems using disambiguation

I think of disambiguation as the algebraic notation oddity that can be used to escape the restrictive set of only 840 possible mating moves mentioned above, so let's look at it in greater detail. Disambiguation is necessary when multiple pieces of the same type can move to the same square, and is done by specifying the file and/or rank of departure of the moving piece. Both FIDE and PGN say that (1) file specification has priority over rank specification. The PGN standard is more thorough and adds that (2) disambiguation is done even if the presence or absence of a check or checkmate symbol already makes the move unique. It also clarifies that (3) disambiguation isn't used to distinguish from moves that aren't legal because of a pin.

The priority rule (1) is the most important one, and it has precise implications for solvers: a move with a file specification like Sdc6 implies that a second knight could have moved to c6 but not from the d-file. A move with a rank specification like S4c6 implies that a second knight could have moved to c6 from the same file as the knight that actually moved.

Length	Without disambiguation	With disambiguation
3.0	3...Qd4#
3.5
4.0	4...b5# 4...Qb5#
4.5	5.Sg3# 5.Qxe4#
5.0	5...Rh1#	5...S4c6# 5...S8h6# 5...Sdc6#
5.5	6.gxf8=S# (1)
6.0
6.5	7.Ka3# (2)	7.B2g5# (3) 7.Q1xc7# (4)
7.0	7...Kxb7# (5)	7...Q1c1# (6) 7...Q8xb2# (7)
7.5		8.Qfa6# (8) 8.Qaxc2# (9)

Table

The problems in 5.0 moves or fewer were found by a computer search that targeted all SAN moves, irrespective of disambiguation. The five originals plus my Retros column original were found by a computer search specifically tailored to find disambiguating checkmate problems. The suggestion of using disambiguation to beat length records came from Juha Saukkola, whom I thank in passing.

Solutions to the game-determining move problems

1.e4 d5 2.exd5 Sc6 3.dxc6 a5 4.cxb7 Kd7 5.bxc8=Q+ Kc6 6.Qf3+ Kb6 7.Qf6+ Ka7 8.Qfa6#