What is HAP notation?

This is a kind of chess game notation, suitable for problems only, invented by Mark Tilford during his sleep (!) as is related here: www.greylabyrinth.com/discussion/viewtopic.php?t=11609.

Peter Fayers gave an excellent definition of it in Variant Chess 55 (2007/09):
"You are given a game score, but instead of being told which specific units move and the coordinates of the displacement, your are only told the type of unit. This can either be Human (K, Q or B), Animal (N or R, harking back the early days of chess when the Rook was the elephant), or Pawn (which, being mere foot-soldiers, are not considered human, a trait followed by the officer class over the centuries). Captures (and type captured) and checks are noted."

To this we'll add that castling is noted O (but without more precision, so we don't know if it's short or long range castling), promotion is noted P=A or P=H, en passant capture is noted PxP, no difference is made between check types (simple, discovered, double...), mate is noted #, stalemate, dead position or draw (50 moves, threefold repetition) are noted =.

It's obvious that every chess game can be written in HAP notation, but often different games give the same HAP translation. For example 1.P P 2.P H# (the fool's mate) stands for 8 different games: 1.g4 e6 2.f3 Qh4#, 1.f4 e5 2.g4 Qh4#... So the idea is to find a HAP notation which belongs to only one game, for example 1.P P 2.PxP P 3.H# which is a simple variant of the above, and ask the solver to recover the original game.

The whole theme recalls Éric Angelini's StenoChess notation, which is also problem oriented. You can read more about it here: http://www.cetteadressecomportecinquantesignes.com/Steno.htm.

HAP Problems

The problems are sorted by date, not by difficulty. Generally the shorter the game is, the easier it is to solve. A C+ symbol means it has been computer tested, and unless otherwise stated Mario Richter kindly made the testing.

At first Mario used his homebrew retroanalysis program to test all problems from the GreyLabyrinth's forum (see address above), but only one was leading to a unique position, though the order of moves is not uniquely fixed:
1) Find the unique position resulting from a game written 1.A A 2.A A 3.AxP A 4.AxA A 5.AxA A 6.AxA A 7.AxA (Russell Ernest Rice, 2006/12, C+).

Now to originals, uniquely determined games:
2) Find the unique game written 1.P P 2.PxP A 3.AxA HxA 4.P HxP 5.PxP HxH 6.P=A H 7.AxA H 8.AxH+ H 9.AxH# (Alain Brobecker, 2007/10, C+).

Bernd Gräfrath produced two very pleasant mate problems containing castling. The first one i was able to recover without chessboard during a dull work reunion, thank you Bernd!
3) Find the only game that is noted 1.P P 2.A A 3.H H 4.AxP AxP 5.O HxP# (Bernd Gräfrath, 2007/11/23, C+).

The second one is logical, but required a chessboard to solve.
4) Find the unique game written 1.P P 2.H H+ 3.H PxP 4.A PxA 5.O PxH+ 6.H AxH 7.A P=A 8.H A 9.AxA HxA# (Bernd Gräfrath, 2007/11/23, C+).

This one is just a version of an existing problem. Will you recognise the theme?
5) Find the unique game written 1.P P 2.PxP HxP 3.H HxP 4.HxP HxA 5.AxP AxH 6.AxP HxH 7.AxP HxP 8.AxP+ HxA 9.HxH AxP 10.HxH AxP 11.HxA AxA+ 12.AxA AxH 13.AxP AxP 14.AxA AxP 15.AxH AxP 16.AxP+ HxA 17.HxA= (Samuel Loyd, NY Clipper, 1895, version 2007/11/24, C+ by AB using François Labelle's work).

Another problem by Bernd, this time including a special capture...
6) Find the unique game written 1.P P 2.P P 3.H+ P 4.PxP+ H 5.P P 6.H PxP+ 7.H P=A 8.PxH=A# (Bernd Gräfrath, 2007/11/24, C+).

The next one was designed primarily for its first move...
7) Find the unique game written 1.P P 2.H P 3.HxP+ H 4.A PxP 5.H+ H 6.O PxP+ 7.AxP# (Alain Brobecker, 2007/11/28, C+ and H+ by MR).

The same day Bernd Gräfrath & Joost de Heer were submitting games starting with the same move. Joost's problem was very short (but François Labelle gave a shorter one), while Bernd's shows two castlings.
8a) Find the unique game written 1.P P 2.H HxH 3.H HxH 4.P HxP+ 5.HxH PxP (Joost De Heer, 2007/11/30, C+).
8b) Find the unique game written 1.P P 2.H P 3.HxH P 4.HxP PxP+ 5.AxP A 6.HxA O 7.H H 8.H A 9.O HxH+ 10.H HxH+ 11.HxH AxP (Bernd Gräfrath, 2007/11/30, C+).

Bernd shows a nice, usual, theme:
9) Find the unique game written 1.P P 2.P P 3.H+ P 4.PxP+ H 5.P A 6.P=A H 7.A P 8.PxP O (Bernd Gräfrath, 2007/11/30, C+).

Another classical theme:
10) Find the unique game written 1.P P 2.P P 3.P P 4.PxP PxP+ 5.AxP H 6.PxH=H A 7.AxP AxA 8.P AxH 9.P A 10.P O 11.PxH P 12.PxA=A P 13.AxP HxA 14.P P 15.P P 16.P P=A 17.P AxH 18.P=H AxH+ 19.AxA H 20.HxH+ (Bernd Gräfrath, 2007/12/01, C+).

François Labelle proposed an infinite game. He considers that no draw is claimed by any player under the threefold position repetion or the 50 moves rules (ie normally the Codex for chess composition says that if a draw can be claimed it must be, so François adds the "No Codex" stipulation).
11) Find the unique game written 1.A A 2.A A 3.AxP AxP 4.AxA AxA 5.A A 6.AxP AxP 7.AxA AxA 8.A A 9.HxA HxA 10.A A 11.A A 12.A A 13.A A ...... n.A A ...... (François Labelle, 2007/12/01, C+ by FL).

Bernd pushed the length limit further (with the Codex), using the same theme and start as in his problem 10.
12) Find the unique game written 1.P P 2.P P 3.P P 4.PxP PxP+ 5.AxP H 6.PxH=H A 7.AxP AxA 8.P AxH 9.P A 10.P O 11.PxH P 12.PxA=A P 13.AxP HxA 14.P P 15.P P 16.P P=A 17.P AxH 18.P=H AxH+ 19.AxA H 20.HxH+ A 21.HxA+ H 22.H+ PxH 23.P H 24.P H 25.PxP H 26.P H 27.A H 28.A H 29.PxA=A HxP 30.AxA H 31.P P 32.P P 33.P P 34.P P 35.P=A P=A 36.AxP A# (Bernd Gräfrath, 2007/12/02, C+).

François found an amusing quadruple:
13) Find the unique game starting with 1.P P 2.P H 3.PxH H 4.PxP HxA and which can finish as 5.P=A, 5.P=H, 5.P=A+ or 5.P=H# (François Labelle, 2007/12/03, C+ by FL).

Russel Rice proposes a theme classical in proof games.
14) Find the unique game written 1.P P 2.PxP H 3.P HxP 4.PxP A 5.PxH=A+ A 6.AxA HxA 7.AxP HxA 8.AxH (Russell Ernest Rice, 2007/12/07, C+).

And again the same theme, but with another type of piece.
15) Find the unique game written 1.P P 2.P H 3.P HxA 4.PxP HxA 5.PxH=A A 6.AxP A 7.AxP AxP 8.A AxA 9.AxH (Russell Ernest Rice, 2007/12/07, C+).

Determined first moves & statistics

One can wonder if it's possible to find a game starting with a given white move, and which is uniquely determined by its HAP notation. Moreover, if that is possible, what is then the shortest game starting with a given move?

Joost de Heer and i made some research, but then François Labelle modified his homebrew retroanalysis program to compute every possible HAP string. He first gave the results up to ply 7, and since only 1.a3 was missing Joost had time to provide a valid candidate for this move. It's only after 2.5 days of computation that François gave the results up to ply 8. You can download an archive containing the output of François' program here: hap.zip. Since the extensive search allows for some statistics, here they are:

ply	1	2	3	4	5	6	7	8
number of possible HAP strings	2	4	15	75	468	3152	23,500	184,462
number of HAP strings that uniquely determine a game	0	0	0	1	5	59	341	2,652
number of problems ending in checkmate	0	0	0	0	1	5	25	167
number of problems with castling	0	0	0	0	0	0	10	170

The 8-ply HAP string with the most solutions is 1.P P 2.P P 3.P P 4.P P (1,779,740,200 solutions). François also pushed the computation up to ply 9 for white P moves, and can say that there are 88 problems in 4.5 moves ending with a promotion. See problem 13 for a nice example. Last, here's one of the shortest problem with castling: 1.A P 2.P H 3.HxP HxA 4.O (François Labelle, 2007/11/30, C+).

François also discovered there's no problem with an en passant capture in 8 plies or less. So the question is: What is the shortest game uniquely written in HAP notation with an en-passant capture? See problem 6 for an example.

Back to the original problem. Below is a table containing all the shortest HAP strings that uniquely determine a game and starting with a given white's first move. Unless otherwise stated, the problems have been provided and computer tested by François Labelle on 2007/11/30. A star indicates that there's no other solution with the same number of moves.

The next question asked by Mario concerns the possibility to find an HAP string for every black's first move. François also quickly answered, except for one problem that required 8 plies and more computing. Unless otherwise stated, the problems have been provided and computer tested by François Labelle on 2007/12/04. A star indicates that there's no other solution with the same number of moves.

Extension of the theme

It's possible to extend the theme (forgetting the coordinates) to other sets of pieces. I think captures, checks, mates, draws and promotions shall be noted as usual, but before a problem one could give the sets used. For example:

• Human={K;Q;B}; Animals={R;N}; Pawns={P}; O={o-o;o-o-o} is the definition of the HAP notation given before. Bernd proposed to call it OHAP.

• Human={K;Q;B;o-o;o-o-o}; Animal={R;N}; Pawn={P} is a "harder" variant of the normal notation, called EHAP, where castling is noted only with a H. The following problems (where you're asked to "Find the unique game written ...") are all valid, computer tested, versions of the previous ones.

  1. 1.P P 2.H H+ 3.H PxP 4.A PxA 5.H PxH+ 6.H AxH 7.A P=A 8.H A 9.AxA HxA#
  2. 1.P P 2.H P 3.HxP+ H 4.A PxP 5.H+ H 6.H PxP+ 7.AxP#
  3. 1.P P 2.P P 3.H+ P 4.PxP+ H 5.P A 6.P=A P 7.PxP H 8.AxH H 9.HxH+
  4. 1.P P 2.P P 3.P P 4.PxP PxP+ 5.AxP H 6.PxH=H A 7.AxP AxA 8.P AxH 9.P A 10.P H 11.PxH P 12.PxA=A P 13.AxP HxA 14.P P 15.P P 16.P P=A 17.P AxH 18.P=H AxH+ 19.AxA H 20.HxH+
  5. 1.P P 2.P P 3.P P 4.PxP PxP+ 5.AxP H 6.PxH=H A 7.AxP AxA 8.P AxH 9.P A 10.P H 11.PxH P 12.PxA=A P 13.AxP HxA 14.P P 15.P P 16.P P=A 17.P AxH 18.P=H AxH+ 19.AxA H 20.HxH+ A 21.HxA+ H 22.H+ PxH 23.P H 24.P H 25.PxP H 26.P H 27.A H 28.A H 29.PxA=A HxP 30.AxA H 31.P P 32.P P 33.P P 34.P P 35.P=A P=A 36.AxP A#
  Also Bernd send some problems specifically created for EHAP:
  998) Find the unique game written 1. P P 2. P P 3. H+ P 4. PxP+ H 5. P A 6. P=A P 7. PxP H 8. AxH H 9. HxH+ (Bernd Gräfrath, 2007/12/06, C+).
  999) Find the unique game written 1.P P 2.H P 3.HxH P 4.HxP PxP+ 5.AxP A 6.A AxA 7.H H 8.HxP HxH 9.HxH AxH 10.H H# (Bernd Gräfrath, 2007/12/12, C+).
• Royal={K;Q}; Noble={R;N;B}; Peasant={P}; O={o-o;o-o-o} would be a different theme, probably allowing some interesting problems too.
• It would be possible to include a same piece type in more than one set, thus having more ambiguities and more difficult problems.

Solutions

1) Total elimination of the black Animals. Position after, for example:
1.Nc3 Nh6 2.Nd5 Nf5 3.Nxe7 Rg8 4.Nxg8 Ne7 5.Nxe7 Nc6 6.Nxc6 Rb8 7.Nxb8

2) Switchback of a piece which is then captured on its home square.
1.a4 b5 2.axb5 Na6 3.Rxa6 Bxa6 4.b6 Bxe2 5.bxc7 Bxf1 6.c8=R Ba6 7.Rxa8 Qb8 8.Rxb8+ Bc8 9.Rxc8#

3) Short range castling.
1.g4 e5 2.Nf3 Nh6 3.Bg2 Qh4 4.Nxe5 Nxg4 5.o-o Qxh2#

4) And of course the long range one.
1.d4 c5 2.Bh6 Qa5+ 3.Qd2 cxd4 4.Nc3 dxc3 5.o-o-o cxd2+ 6.Kb1 Nxh6 7.Rc1 d1=R 8.Ka1 Re1 9.Rxe1 Qxe1#

5) A massacre proof game, all 30 pieces are captured in the minimum of 33 plies! François Labelle used a homebrew program to prove there are 25 such games finishing with wKf2+bKf7 and 866 games finishing with wKf2+bKe7 and nothing else. All i had to do was to list them (with Popeye), convert them in HAP notation and check for uniqueness. 745 games out of 891 have a unique HAP notation. You can find all of them here: hap.zip.
1.e4 d5 2.exd5 Qxd5 3.Bd3 Qxa2 4.Bxh7 Qxb1 5.Rxa7 Rxh7 6.Rxb7 Qxc1 7.Rxc7 Qxc2 8.Rxe7+ Kxe7 9.Qxc2 Rxh2 10.Qxc8 Rxg2 11.Qxb8 Rxg1+ 12.Rxg1 Rxb8 13.Rxg7 Rxb2 14.Rxg8 Rxd2 15.Rxf8 Rxf2 16.Rxf7+ Kxf7 17.Kxf2=

6) En passant capture.
1.c4 d5 2.c5 d4 3.Qa4+ b5 4.cxb6ep+ Qd7 5.b7 d3 6.Kd1 dxe2+ 7.Kc2 e1=R 8.bxc8=R#

7) Single step pawn move. The problem is H+ because Mario Richter gave the following proof for it:
* If a pawn move that gives check (P+ or Px?+) is answered by a capture of a pawn, then it's the check giving pawn that is captured and the king was directly attacked by the pawn. So 6.O P3xP2+ 7.AxP2#.
* Black made 4 pawns move, so 1.P P5 2.H P4 3.HxP7+ H 4.A P4xP3 5.H+ H 6.O P3xP2+ 7.AxP2#, and we can also conclude that 1.P was on the third rank.
* 3.HxP7+ can't be 3.Qxe7+ since the only answer would be Kxe7. So the check 3.HxP7+ was given diagonally, either 3.Hxd7+ Kf7 or 3.Hxf7+ Kd7. To reach d7 or f7, two H move by white (wQ or wBf1) were necessary.
* After 3... Kf7 or 3... Kd7 the K is standing on a light square, so wBc1 has not moved, and thus O is short range castling, and we have: 1.P3 P5 2.H P4 3.HxP7+ K7 4.Ng1- P4xP3 5.H+ H 6.o-o P3xP2+ 7.AxP2#
* In 7.AxP2# it's not possible to have discovered check by a knight, so 7.Rf1xPf2#, and we now know that: 1.P3 Pf7-f5 2.H Pf5-f4 3.HxP7+ Ke8-f7 4.Ng1- Pf4xP3 5.H+ Kf7-f6 6.o-o P3xPf2+ 7.Rf1xf2#
* bKf6 has the flight square e5. The only way to control this is with wQh5, so the complete game is now:
1.e3 f5 2.Bb5 f4 3.Bxd7+ Kf7 4.Nh3 fxe3 5.Qh5+ Kf6 6.o-o exf2+ 7.Rxf2#

8a) A very nice setup.
1.b3 e5 2.Ba3 Bxa3 3.Qc1 Bxc1 4.f4 Bxd2+ 5.Kf2 exf4

8b) Short and long range castling.
1.b3 e5 2.Ba3 e4 3.Bxf8 e3 4.Bxg7 exd2+ 5.Nxd2 Nf6 6.Bxf6 o-o 7.Qc1 Qe7 8.Qa3 Re8 9.o-o-o Qxa3+ 10.Bb2 Qxb2+ 11.Kxb2 Rxe2

9) Valladao-Monteiro (ep+castling+promotion) task:
1.c4 d5 2.c5 d4 3.Qa4+ b5 4.c5xb6e.p.+ Qd7 5.b7 Nc6 6.b8N Bb7 7.Na6 d3 8.exd3 0-0-0

10) All 4 promotions (Allumwandlung=AUW)
1.h4 c5 2.h5 c4 3.h6 c3 4.hxg7 cxd2+ 5.Nxd2 Qc7 6.gxf8B Nf6 7.Rxh7 Nxh7 8.b4 Nxf8 9.b5 Nh7! 10.b6 0-0 11.bxc7 b5 12.cxb8N b4 13.Nxd7 Bxd7 14.c4 b3 15.c5 b2 16.c6 b1R 17.c7 Rxc1 18.c8Q Rxd1+ 19.Rxd1 Bg4 20.Qxg4+

11) Elimination of all Animals but two rooks that are caged.
1.Nc3 Nc6 2.Nd5 Nd4 3.Nxe7 Nxe2 4.Nxg8 Nxg1 5.Nh6 Nh3 6.Nxf7 Nxf2 7.Nxh8 Nxh1 8.Nf7 Nf2 9.Kxf2 Kxf7 10.Rb1 Rb8 11.Ra1 Ra8 ...

12) All 4 promotions (AUW), smothered mate.
1.h4 c5 2.h5 c4 3.h6 c3 4.hxg7 cxd2+ 5.Nxd2 Qc7 6.gxf8B Nf6 7.Rxh7 Nxh7 8.b4 Nxf8 9.b5 Nh7! 10.b6 0-0 11.bxc7 b5 12.cxb8N b4 13.Nxd7 Bxd7 14.c4 b3 15.c5 b2 16.c6 b1R 17.c7 Rxc1 18.c8Q Rxd1+ 19.Rxd1 Bg4 20.Qxg4+ Ng5 21.Qxg5+ Kh7 22.Qg6+ fxg6 23.f4 Kg7 24.f5 Kf6 25.fxg6 Ke5 26.g7 Kd4 27.Nh3 Kc3 28.Nf2 Kb2 29.gxf8R Kxa2 30.Rxa8 Kb2 31.g4 a5 32.g5 a4 33.g6 a3 34.g7 a2 35.g8N a1N 36.Nxe7 Nc2#

13) Nice little problem showing all promotions (AUW) in different lines of play.
1.c4 d5 2.c5 Qd6 3.cxd6 Bf5 4.dxc7 Bxb1 then 5.c8=N, 5.c8=B, 5.c8=R+ or 5.c8=Q#

14) Phoenix-style promotion.
1.b4 c5 2.bxc Qa5 3.c6 Qxa2 4.cxb Nc6 5.bxc8=R+ Nd8 6.Rxa8 Qxb1 7.Rxa7 Qxa1 8.Rxa1

15) Phoenix-style promotion.
1.h4 e5 2.h5 Qh4 3.h6 Qxh1 4.hxg Qxg1 5.gxf8=N Nf6 6.Nxd7 Ne4 7.Nxe5 Nxd2 8.Nf3 Nxb1 9.Nxg1

The shortest HAP problems starting with a given white's first move...
1.a3 Nc6 2.Nf3 e5 3.Nxe5 Bxa3 4.Nxc6 Bxb2
1.a4 b5 2.Ra3 Ba6 3.Rb3 bxa4 4.Rxb8
1.b3 e6 2.Ba3 Qf6 3.Bxf8 Qxa1
1.b4 Nf6 2.Bb2 e6 3.Bxf6 Bxb4
1.c3 d5 2.Qb3 Bf5 3.Qxd5 Bxb1 4.Qxd8+
1.c4 a5 2.Nc3 a4 3.Nxa4 Rxa4 4.Qxa4
1.d3 g5 2.Nc3 Bg7 3.Bxg5 Bxc3+
1.d4 Nc6 2.Qd3 Nxd4 3.Qf3 Nxf3+
1.e3 b5 2.Nf3 Bb7 3.Bxb5 Bxf3
1.e4 f5 2.exf5 g5 3.Qh5#
1.f3 e5 2.Kf2 Qh4+ 3.Ke3 Qd4#
1.f4 e5 2.Nf3 Qh4+ 3.Nxh4 exf4
1.g3 d5 2.Bh3 Qd6 3.Bxc8 Qxg3 4.Bd7+
1.g4 d6 2.Bg2 Bxg4 3.Bc6+
1.h3 d5 2.Nc3 Bxh3 3.Nxd5 Qxd5 4.gxh3
1.h4 g5 2.Rh3 Bh6 3.Rg3 gxh4 4.Rxg8+
1.Na3 e5 2.d4 exd4 3.Qxd4 Bxa3
1.Nc3 d5 2.Nxd5 Qxd5
1.Nf3 e5 2.Nxe5 Nc6 3.Nxc6
1.Nh3 e5 2.f4 Qf6 3.fxe5 Qxf1+ 4.Rxf1

The shortest HAP problems starting with a given black's first move...
1.e4 a6 2.Bxa6 d5 3.exd5 Qxd5 4.Bxb7
1.c4 a5 2.Nc3 a4 3.Nxa4 Rxa4 4.Qxa4
1.e3 b6 2.Qf3 Ba6 3.Qxa8 Bxf1
1.Nf3 b5 2.e3 Bb7 3.Bxb5 Bxf3
1.d3 c6 2.Bf4 Qb6 3.Bxb8 Qxf2+ 4.Kd2
1.d4 c5 2.Bf4 Qb6 3.Bxb8 cxd4
1.g4 d6 2.Bg2 Bxg4 3.Bc6+
1.Nc3 d5 2.Nxd5 Qxd5
1.b4 e6 2.Bb2 Nf6 3.Bxf6 Bxb4
1.Nf3 e5 2.Nxe5 Nc6 3.Nxc6
1.d4 f6 2.Qd3 Nc6 3.Qxh7 Nxd4 4.Qg6#
1.e4 f5 2.Qf3 fxe4 3.Qxf8+
1.d4 g6 2.Qd3 Bh6 3.Qxg6 Bxc1 4.Qxg8+
1.Nc3 g5 2.d3 Bg7 3.Bxg5 Bxc3+
1.d4 h6 2.Bxh6 f6 3.Qd3 gxh6 4.Qg6#
1.e4 h5 2.Qxh5 Nf6 3.Qxh8 Nxe4
1.e4 Na6 2.Bxa6 d5 3.exd5 Qxd5
1.d4 Nc6 2.Qd3 Nxd4 3.Qf3 Nxf3+
1.b4 Nf6 2.Bb2 e6 3.Bxf6 Bxb4
1.d4 Nh6 2.Bxh6 c5 3.Bxg7 cxd4 4.Bxf8 Qa5+

998) Valladao-Monteiro task in EHAP.
1.c4 d5 2.c5 d4 3.Qa4+ b5 4.cxb6e.p.+ Qd7 5.b7 Nc6 6.b8N d3 7.exd3 Ba6 8.Nxd7 0-0-0 9.Qxa6+

998) Both castlings in EHAP.
1.b3 e5 2.Ba3 e4 3.Bxf8 e3 4.Bxg7 exd2+ 5.Nxd2 Nf6 6.Ne4 Nxe4 7.Bf6 0-0 8.Qxd7 Qxf6 9.Qxc8 Rxc8 10.0-0-0 Qa1#