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Abstract

A 5x5 board is the smallest board on which one can set up all kind of chess pieces as a start position. We consider Gardner's minichess variant in which all pieces are set as in a standard chessboard (from Rook to King). This game has roughly 9x10^{18} legal positions and is comparable in this respect with checkers. We weakly solve this game, that is we prove its game-theoretic value and give a strategy to draw against best play for White and Black sides. Our approach requires surprisingly small computing power. We give a human readable proof. The way the result is obtained is generic and could be generalized to bigger chess settings or to other games.
arXiv:1307.7118v1 [cs.GT] 26 Jul 2013
Gardner’s Minichess Variant is solved
Mehdi Mhalla
Mehdi.Mhalla@imag.fr
CNRS Universit´e de Grenoble - LIG, B.P. 53
- 38041 Grenoble Cedex 09, France
Fr´ed´eric Prost
Frederic.Prost@imag.fr
Universit´e de Grenoble - LIG, B.P. 53
- 38041 Grenoble Cedex 09, France
July 29, 2013
Abstract
A 5× 5 board is the smallest board on which one can set up all kind of chess pieces as a start position.
We consider Gardner’s minichess variant in which all pieces are set as in a standard chessboard (from
Rook to King). This game has roughly 9 × 10
18
legal positions and is comparable in this respect with
checkers. We weakly solve this game, that is we prove its game-theoretic value and give a strategy to draw
against best play for White and Black sides. Our approach requires surprisingly small computing power.
We give a human readable proof. The way the result is obtained is generic and could be generalized to
bigger chess settings or to other games.
1 Introduction
Solving popular ga mes like Othello, Checker s or Chess tantamount to the grail search in the field of computer
games. The resolution of checkers [SBB
+
07] put a mark in the field in the sense that the space search of
this game is enormous (5 × 10
20
) and the difficulty to make cor rect move decisions fairly high.
The ga me of ches s have always bee n recognized as the ultimate challenge in artificial intelligence. Since the
early days of computer science chess and computers have interacted together [Pro12]. Nowadays computers
have superhuman strength a nd the game is partially solved: endgame databases up to few pieces have
been computed. The most famous ones being the Nalimov ta ble s (6 pieces). Recently Lomonosov endgame
tablebases [Ltd13] have been computed and give perfect play for 7 pieces (the size of the tablebase is 140
Terabytes). Nevertheless, the resolution of chess re mains too difficult to be imagined: the number of leg al
positions is something around 10
45
[All94] and decision complexity is very high (the amount of chess literature
is a proof by itself).
Some studies have been done to resolve particular cases of chess on smaller bo ard. Notably, 3×3, 3×4 and
4×4 (limited to 9 pieces on the board) chess variants have been solved by K. Kryukov [Kry04, Kry09, Kry11].
In these variants there is no starting position as in traditional chess. Positions are treated as puzzles. Each
variant is strongly solved in the sense that the game-theoretic value of all le gal position is determined together
with the perfect play associated. The number of legal positions is roughly 3 × 1 0
15
for the 4 × 4 variant
[Kry11].
In this paper we study the variant called Gardner’s Chess. It is played on a 5 × 5 board, the initial
position is the initial position of chess but for the three pieces on the King side that are removed. The rules
are the ones of classical chess without the two squares move for Pawns, en passant moves and castling. This
variant has roug hly 9 × 10
18
legal positions. This variant has been played extensively notably in Italy by
corres pondence [P ri07]. The results of finished games were the following:
White victory 40%
draw 32%
Black victory 28 %
1
2 Results
The game-theoretic value of Gardner’s Chess is draw. We prove this by giving two or acles, one for White
and one for Black. Both oracles can force draw versus best play. The intersection of the two oracles g ives
flawless games. Thus Gardner’s chess is weakly solved.
The proof is surprisingly small and can be totally checked by a human. Oracles are given in appendix
3 for the White side and appendix 4 for the Black side. From this point of view our result strongly differs
from the resolution of checkers despite the fact that space search and difficulty of decision are of the same
order of magnitude in both ga mes. Indeed, the proof of [SBB
+
07] is not checkable by human eyes: it has
required a n enormous computing power (hundreds of computers in para lle l over a decade).
Most of our work was achieved with consumer-grade laptop computers. We have adapted the open
source Stockfish chess e ngine [RCK
+
10] to play Gardner’s Chess mainly by restricting the movements
to the part of the board and changing the promotion ranks. Sources, executables for several environ-
ments and various files, inc luding the oracles in PGN format as well as the list of the perfect o pen-
ings for Gardner’s Chess, can be found at the author’s Minichess Resolution page: “http://membres-
lig.imag.fr/prost/MiniChessRes olution/”.
The main line of oracles were computed in a semi-automated way: we were mainly following the most
equalizing line. I t turns out that most of the deviations from the main line can be quickly decided. It is
mainly due to the fact that in Gardner’s chess pawns are immediately exchanged or blocked. Moreover,
pieces cannot develop naturally since almost all free squares are controlled by pawns. Also the fact that
promotion ha ppens quickly leads to some very rapid checkmates that allow to prune the game tree.
Using these Oracles it is imp ossible to lose. Oracle for White (resp. for Black) does not examine
alternative choices for White (resp. Black) decis ion nodes but indicates how to answer every possible Black
(resp. White) ”r e asonable” move. Unreasonable moves, i.e. moves that obviously lead to a position where
it is clear that Black (resp. White) cannot win can be dealt with our engine. We provide the maximal
number of moves required to mate for our engine (not necessarily the distance to mate). Nevertheless, in
these positions, from a human point of view, it is easy not to lose.
As a by-product of our study on Gardner’s Chess the analysis of perfect openings shows the positions in
which the evaluation of Stockfish is tricked. Indeed for some positions while showing largely “won” evaluation
(up to +6) the po sition is completely equal. What is inter e sting is that these evaluation bugs ca n be found
on a 8x8 board as well. T hus the analysis of these positions may help to improve the evaluatio n of Stockfish
for standard chess games.
A complete description of the openings in gardner mini chess as well as a sample of tricky draws and
difficult checkmates c an be found at “http://membres-lig.imag.fr/prost/MiniChessResolution/”
3 Gardner : Oracle for White Draw
We give an oracle for the White side of the Gardner variation. The objective is to fo rce a dr aw versus the
best play. Therefore, we give it as a tree of variations that needs no explanations on White nodes: it is
maybe possible to find shorter draw (or even win) but our aim is to have an oracle the most readable from
a human point of view: the definitive judgment on the leafs of this tree are clear.
Since there are no choices to be explored for White nodes we adopt the following convention to name
sub-variations: first we note the depth in the oracle, then we enumerate deviations from the main line by
enumerating Black (relevant) moves from left to right, pawns come first, after we enumerate moves o f the
pieces following the lexical order g oing fr om left to right and top to bottom. Thus the varia tion [3|1.3.7] is
the one obta ine d by following the oracle until depth 3 and selecting as sub-variation move 1 as the firs t move
for B lack, then move 3 as second move for Black and 7 as the third move. We write + (resp. + ) when
it is obvious that B lack (resp. White) cannot win. We write ♯x♯
(resp. ♯x♯
) when there exists a forced
checkmate of the Black King (res. White King) in x moves (though it is poss ible that shorter checkma tes
exist). Very often positions that look lost (because one side has a piece advantage for instance) can be fully
2
decided by our engine as forced checkmates. Justifying lines are written like this: 1 b4 cXb4 2 cXb4 d4.
Finally, the coordinate of the lower left square is b2. Hence the starting position is:
6
snaqj
5
popop
4
Z0Z0Z
3
POPOP
2
SNAQJ
b c d e f
In this position the Black move identified by 1 is . . . b4 and move number 6 is . . . Nb4, move number 7
is . . . Nc4.
We give the White oracle as a variation tree. After each move of the oracle we start by giving all lines in
which a forced checkmate can be found using our engine.
1 b4
(
1. . . d4 2 bXc5 47
after 2. . . BXc5 3 f4 both the pressure on the b file and on diagonal b2 f6
are too strong to be sustained by by Black. 1. . . e4 2 bXc5 28
, the point is that o n 2. . . BXc5 3 d4 the
threat of RXb5 combined with the la ck of spac e for Black is too hard to be met. Other moves just lose a
piece at least. 1. . . f4 2 bXc5 24
2... Bxc5 3.d4 the threat of RXb5. 1. . . NXb4 2 cXb4 24
White
is a piece up for nothing. 1. . . Nd4 2 bXc5 17
White is a piece up for nothing.
):
[1|1] 1. . . c4 2 d4 (
2. . . BXb4 3 dXe5+ 29
, 2. . . Bc5 3 bXc5 8
, 2. . . NXd4 29
, 2. . . f4 3
e4 38
).
[1|1.1] 2. . . eXd4 3 eXd4 (
3. . . Bc5 6
, 3. . . Ne5 14
, 3. . . Qe3+ 6
, 3. . . Qe4 10
,
3. . . Qe5 12
, 3. . . Be531
):
[1|1.1.1] 3. . . f4 4 QXe6+ KXe6 = White just has to move his King on e2-f2 and Black
cannot break through. No matter wha t is the relative position of the two Kings, if black
Knight takes on d4 or b4 White takes back with the Knight and the position is s till blocked
for Black and if Black plays . . . BXb4 the position is 17
when kings on file e and 24
when kings are in file f. Finally, if Black plays . . . Ne5 white just takes it with dXe5 and if
Black plays . . . Be5 White just continue to move his k ing.
[1|1.1.2] 3. . . QXe2+ 4 KXe2 = for the same reason as line [1|1.1 .1].
[1|1.2] 2. . . e4 3 f4 = Black is in zugzwang and must give a piece. Due to the blocked nature
of the position he can do it without losing but he cannot break through e.g. 3. . . Be5 4 f Xe5+
NXe5 5 dXe5+ QXe5 6 Nd4 and White can simply moves back and forth with the Knight.
[1|1.3] 2. . . NXb4 3 dXe5+ (
3. . . BXe5 4 RXb4 20
, 3. . . KXe5 4 cXd4 13
) + :
[1|1.3.1] 3. . . QXe5 4 NXb4 (4. . . BXb4 5 RXb4 25
, 4. . . Qe6 5 e4 17
, 4. . . Qd4 5
eXd4 3
, 4. . . Qe4 5 f Xe4 9
, 4. . . Qf4 5 eXf4 4
, 4. . . QXd3 5 eXf4 4
, 4. . . QXb3
5 BXb3+ 5
, 4. . . d4 5 cXd4 10
, 4. . . f4 5 eXf4 10
, 4. . . Rc6 5 NXc6 10
):
- [1|1.3.1.1 ] 4. . . Bc5 5 Nc2 = White blocks the position on the dark squares with Nd4
and Rb4 (a nd moves his Rook between b2 -b4 if Black does not move. 5. . . f4 6 Nd4 b4
(other moves leads to a loss for Black) 7 eXf4 BXd4+ 8 cXd4 QXe2+ (other moves lead to
direct checkmate).
- [1|1.3.1.2] 4. . . Ke6 5 Nc2 + similar to line [1|1.3.1.1].
3
1. . . cXb4 2 cXb4 All Knight and Bishop moves lose a piece and end up in a position where clearly
Black cannot win (2. . . Nd4 23
, 2. . . NXb4 18
, 2. . . BXb4 15
, 2. . . e4 3 Bc3+ 15
, 2. . . f4 3
bXc5 29
the idea is that the b6 pawn is lost and Black is lacking space 3. . . BXc5 4 d4 B6d5.Rxb5
)
2. . . d4 3 e4 (3. . . f Xe4 4 f Xe4 8
since the threat of Qf3+ is too strong, 3. . . NXb4 15
, 3. . . BXb4
15
, 3. . . Qb3 12
, 3. . . Qc4 8
, 3. . . Qc68
)
3. . . f4 4 BXf4 (
All Black’s alternatives lead to fo rced checkmate since they lose a piece for nothing
4. . . Bc5 8
, 4. . . Qb3 11
, 4. . . Qc4 8
, 4. . . Qd5 5
, 4. . . Qf5 8
, 4. . . NXb4 25
, 4. . . BXb4
25
)
4. . . eXf4 5 Qd2 (5. . . Ne5 6 QXf4+ 2
, 5. . . NXb4 6 NXb4 + d5 cannot be protected and the
threat of QXf4+ forbids 6. . . BXb4)
5. . . Be5 6 Ke2 = White just moves his King on e2-f2 and Black cannot untangle by . . . Ne5 becaus e
of QXf4 and must otherwise give a piece and cannot win.
4 Gardner : Oracles for Black Draw
We give, for each of the White seven legal first move, an oracle from the Black point of view that forces
the draw. So here we give no explanations for Black decision nodes and we explore all reasona ble moves (as
explained erlier) at White decision nodes. These oracles are sometimes much simpler than the White oracle
for draw since, rather curiously, it is sometimes more difficult for White to achieve draw. It means that often
even slight deviations from the main line dire ctly lea d to positions that can be decided as forced checkmates.
4.1 White moves b4
1 b4 cXb4 (
2 Rb3 d4 17
, 2 RXb4 NXb4 21
, 2 c4 bXc4 15
the b4 c4 pawn duo is too strong, 2
Nd4 bXc3 20
, 2 e4 bXc3 25
)
[1|1] 2 d4 bXc3 (
3 dXe5+ 17
, 3 e4 9
, 3 f4 10
, 3 Rb3 17
, 3 Rb4 12
, 3 RXb5 12
, 3
Na4 9
, 3 Qd3 10
, 3 Qc4 10
, 3 QXb5 7
)
3 BXc3 b4 (
4 e4 dXe4 19
, 4 f4 eXf4 30
,4 NXb4 NXb4 26
, 4 BXb4 NXb4 24
, 4 Bd2
b3 23
, 4 Qd2 bXc3 18
, 4 Qd3 e4 26
, 4 Qc4 dXc4 10
, 4 Qb5 RXb5 9
)
[1|1.1] 4 dXe5+ BXe5 (
5 e4 13
, 5 f4 15
, 5 Rb3 15
, 5 NXb4 22
, 5 Nd4 10
, 5
Qd2 8
, 5 Qd3 15
, 5 Qc4 10
, 5 Qb5 8
, 5 BXb4 17
, 5 Bd2 11
, 5 Bd4 29
,
5 BXe5+ QXe5 39
)
5 RXb4 RXb4
(
6 e4 9
, 6 f4 9
, 6 Bb2 8
, 6 Bd2 10
, 6 Bd4 9
, 6 BXe5+ 51?
, 6 NXb4 27
, 6
Nd4 12
, 6 Qd2 1 1
, 6 Qd3 20
, 6 Qc4 6
, 6 Qb5 7
)
6 BXb4 NXb4 (
7 e47
, 7 Nd414
, 7 Qb59
, 7 Qc46
, 7 Qd36
, 7 Qd217
)
[1|1.1.1] 7 f4 Bc3 (
8 e4 8
, 8 Nd4 13
, 8 Qb5 10
, 8 Qc4 5
, 8 Qd3 5
,
8 Qd2 4
, 8 Qe2 8
, 8 Ke2 7
) 8 NXb4 d4 (9 eXd4 18
, 9 Nd5 17
, 9
Qd3 16
, 9 Qf3 16
, 9 Nc6 14
, 9 Kf3 10
, 9 e4 7
, 9 Qd2 5
, 9 Qb2 QXe3
checkmate, 9 Qc2 QXe3 checkmate, 9 Qc4 QXe3 checkmate, 9 Qb5 QXe3 checkmate)
= since on both Nc2 and Nd3 black exchanges Queen on e3 and the remaining p osition is
draw 9 Nc2 dXe3+ 10 NXe3 Bd4 11 Qe2 BXe3+ 12 QXe3 QXe3+ 13 KXe3.
[1|1.1.2] 7 NXb4 d4 = the only move to avoid . . . dXe3+ and the liquidation of all pawns is
8 e3 Qb3 9 Nd5+ Ke6 10 eXf5+ KXd5 11 f4 Qe3+ 12 QXe3 dXe3+ 13 KXe3.
[1|1.2] 4 Rb3 f4 (
5 e4 19
, 5 Rb2 33
, 5 RXb4 27
, 5 BXb4 22
, 5 NXb4 27
, 5 Qb5
10
, 5 Qc4 8
, 5 Qd2 20
)
4
[1|1.2.1] 5 dXe5+ BXe5 (6 e4 9
, 6 eXf4 17
, 6 Rb2 11
, 6 RXb4 32
, 6 BXb4
39
, 6 Bb2 22
, 6 Bd2 21
, 6 BXe5+ 60
, 6 Nd4 9
, 6 Qd2 14
, 6 Qd3 12
,
6 Qc4 7
, 6 Qb5 1 0
, 6 NXb4 20
)
6 Bd4 NXd4 (7 NXd4 25
, 7 Qd2 9
, 7 Qd3 9
, 7 NXb4 9
, 7 eXf4 8
, 7 RXb4
7
, 7 Rb2 7
, 7 Rd3 7
, 7 e4 7
, 7 Qb5 7
, 7 Rc3 6
, 7 Qc4 6
)
7 eXd4 Bd6 (
8 Qd3 26
, 8 Rb2 23
, 8 Qd2 22
, 8 NXb4 18
, 8 RXb4 13
, 8
Ne3 11
, 8 Rc3 10
, 8 Qe3 7
, 8 Qe4 6
, 8 Qe5 7
, 8 Rd3 10
, 8 Re3 8
, 8
Qb5 6
)
8 QXe6+ KXe6 = White is blocked by Black pawns and canno t pr ogress. The Black King
may just move on f6 f5 squares.
[1|1.2.2] 5 eXf4 eXf4 = 6 Qd3 Qf5 7 Qe2 Qe6 with repetition. White cannot play
Qd2 due to . . . bXc3. Once queen have been exchanged the position is blocked and Bla ck c an
just move his King ad lib.
[1|1.2.3] 5 Bb2 f Xe3+ 6 QXe3 eXd4 7 BXd4+ eXd4 8 QXd4+ Be5 = Black can play
. . . Bc3 on any queen moves and blocks the positio n.
[1|1.2.4] 5 Bd2 eXd4 6 eXd4 QXe2+ 7 KXe2 = White cannot untangle and Black may just
move his King around.
[1|1.2.5] 5 Qd3 eXd4 = similar as line [1|2.8.4.1].
[1|1.3] 4 RXb4 RXb4 = 5 dXe5+ BXe5 6 BXb4 NXb4 7 NXb4 d4.
[1|2] 2 f4 bXc3 (
3 d4 10
, 3 e4 9
, 3 f Xe5+ BXe5 15
, 3 Rb3 15
, 3 Rb4 15
, 3 RXb5
14
, 3 Nb4 10
, 3 Nd4 14
, 3 Qf3 8
, 3 Kf3 8
)
3 BXc3 b4 (4 d4 33
, 4 e4 12
, 4 BXb4 2 6
, 4 Bd4 22
, 4 Bd2 eXf4 23
, 4 Kf3 22
,
4 Qd2 22
, 4 Nd4 19
)
[1|2.1] 4 f Xe5+ BXe5 (
5 e4 12
, 5 Rb3 19
, 5 RXb4 27
, 5 NXb4 26
, 5 Nd4 10
,
5 BXb4 14
, 5 Bd4 23
, 5 Bd2 11
, 5 Qd2 14
, 5 Qf3 12
, 5 Kf3 11
)
[1|2.1.1] 5 BXe5+ QXe5 (6 e4 10
, 6 RXb4 12
, 6 NXb4 15
, 6 Nd4 14
, 6 Qd2
8
, 6 Qf3 8
, 6 Kf3 8
)
- [1|2.1.1.1] 6 Rb3 d4 (
7 Qf3 20
, 7 eXd4 17
, 7 Qd2 14
, 7 e4 13
, 7 Kf3 12
,
7 Rb2 12
, 7 NXb4 11
, 7 RXb4 11
, 7 Rc3 8
)
7 NXd4 NXd4 (8 Qb2 9
, 8 RXb4 8
, 8 Qd2 7
, 8 Rb2 7
, 8 Qc2 5
, 8 Rc3
4
, 8 e4 2
)
8 eXd4 QXd4+ (9 Kf3 12
)
9 Qe3 QXe3+ 10 KXe3 Ke5 + Black can easily achieve draw since the White Rook has
to keep an eye on the b pawn and the Black King is in front of the White d pawn.
- [1|2.1.1.2] 6 d4 Qe4 7 Rb3 f4 = since the only moves that not lose for White ar e either 8
eXf4 QXe2+ 9 KXe2 Kf5 and Black a nd White King move ad lib., or 8 Qd2 f3 9 NXb4
RXb4 10 RXb4 NXb4 11 QXb4 Qc2+ and perpetual check o r 12 Qc5 Qb3 13 Qd6
QXe3+ 14 KXe3 stalema te.
[1|2.1.2] 5 d4 Bd6 (6 e4 15
, 6 NXb4 26
, 6 BXb4 30
, 6 Bd2 45
, 6 RXb4
14
, 6 Qd2 21
, 6 Qd3 24
, 6 Qc4 18
, 6 Qb5 8
, 6 Qf3 32
, 6 Kf3 16
)
6 Rb3 Qe4 + since White cannot do a nything to untangle and Black may just move his
Rook on b5 b6.
[1|2.2] 4 Rb3 d4 (
5 e4 10
, 5 Rb2 10
, 5 RXb4 20
, 5 NXb4 13
, 5 NXd4 14
, 5
Bb2 14
, 5 BXb4 15
, 5 BXd4 14
, 5 Bd2 11
, 5 Qd2 11
, 5 Qf3 10
, 5 Kf3 9
)
[1|2.2.1] 5 eXd4 eXd4 6 QXe6+, other White moves lose s traightforwardly since the Bis op
is lost, 6. . . KXe6 7 BXd4, otherwise Black just moves his Rook on b5 b6 and White cannot
break through 7. . . NXd4 8 NXd4+ Kd5 = the f4 pawn is going to fall and White c annot
win this position.
5
[1|2.2.2] 5 f Xe5+ BXe5 6 RXb4 other moves lose the Rook and lead to q uick White defeat
6. . . RXb4 7 BXb4 NXb4 8 NXb4 dXe3+ 9 QXe3 Qd6 10 Nc2 Qc6 = the best for
White is to repeat moves with 11 Nb4 Qd6.
[1|2.3] 4 RXb4 RXb4 (
5 d4 2 0
, 5 e4 12
, 5 Nd4 12
, 5 BXb4 20
, 5 Bb2 9
, 5 Bd4
14
, 5 Bd2 11
, 5 BXe5+ 18
, 5 Qd2 1 6
, 5 Qf3 12
, 5 Kf3 12
)
[1|2.3.1] 5 f Xe5+ BXe5 6 BXb4 NXb4 7 NXb4 d4 = last pawns will soon be exchanged
and White cannot force any advantage 8 e4 Qb3 9 Nd5+ Ke6 10 Qf3 f Xe4.
[1|2.3.2] 5 NXb4 NXb4 6 f Xe5+ BXe5 7 BXb4 d4 = for the same reas ons as line [1|4.3.1].
[1|2.4] 4 BXe5+ BXe5 (
5 d4 24
, 5 e4 10
, 5 Rb3 21
, 5 RXb4 13
, 5 NXb4 14
, 5
Nd4 14
, 5 Qd2 9
, 5 Qf3 9
, 5 Kf3 9
)
5 f Xe5+ QXe5 6 d4 Qe4 = the position is blocked and Black can just move his King to e6 f6
White can’t remove his Rook from the b file and if he tries to break through the ending will be a
clear draw.
[1|2.5] 4 NXb4 RXb4 (
5 d4 20
, 5 e4 12
, 5 Rb3 10
, 5 Rc2 24
, 5 Rd2 21
, 5 BXb4
31
, 5 Bd2 9
, 5 Bd4 12
, 5 BXe5+ 18
, 5 Qc2 20
, 5 Qd2 17
, 5 Qf3 22
, 5
Kf3 21
)
[1|2.5.1] 5 f Xe5+ BXe5 6 RXb4 NXb4 7 BXb4 d4 = this endgame is completly draw since
a couple of pawns will be ex canged and the remaing ones are mutually blocked.
[1|2.5.2] 5 RXb4 BXb4 6 f Xe5+ NXe5 7 BXb4 Qb6 = .
[1|2.6] 4 Qf3 d4 (
5 eXd4 20
, 5 e4 10
, 5 Rb3 10
, 5 RXb4 20
, 5 NXb4 10
, 5 NXd4
10
, 5 BXb4 19
, 5 Bd2 22
, 5 BXd4 15
, 5 Qe2 10
, 5 Qe4 8
, 5 Qd5 9
, 5
QXc6 11
, 5 Ke2 9
)
5 f Xe5+ NXe5 6 BXd4 b3 + 7 Ke2 bXc2Q+ 8 RXc2 Qb3 White can hold the balance
due to the pin on the Knight and of the threat e4 which forces Black to move back his Queen to
e6.
[1|3] 2 NXb4 NXb4 (
3 c4 bXc4 12
, 3 e4 Nc6 33
, 3 f4 Nc6 33
, 3 Rb3 d4 18
, 3
RXb4 BXb4 25
, 3 Rc2 NXc2 11
)
[1|3.1] 3 cXb4 d4 (
4 Rb3 QXb3 12
, 4 Rc2 Qb3 16
, 4 Bc3 dXc3 1 1
)
[1|3.1.1] 4 eXd4 eXd4 (5 Rb3 9
, 5 Rc2 13
, 5 Bc3 11
, 5 Be3 29
, 5 Bf4 15
,
5 Qe3 8
, 5 Qe4 10
, 5 Qe5 7
)
· [1|3.1.1.1]5 f4 QXe2+ 6 KXe2 Rc6 = the po sition is totally blocked on dark squa rres
and White can only play his King or his Rook on b2 b3.
· [1|3.1.1.2] 5 QXe6+ KXe6 = for the s ame reasons as line [1|3.1.1].
[1|3.1.2] 4 e4 f4 (
5 Rb3 6
, 5 Rc2 16
, 5 Bc3 8
, 5 Be3 10
, 5 Qe3 9
)
the only move leads to a type of drawn position already see n in line [1|1.2] of the White ora cle
5 BXf4 eXf4 6 Qc4 Qe5 = White can just move his Rook on b2-b3 or his King other over
moves are loo sing (he cannot give up the control of the c file).
[1|3.1.3] 4 f4 eXf4 (5 e4 29
, 5 Rb3 1 2
, 5 Rc2 17
, 5 Bc3 2
, 5 Qf3 19
, 5 Kf3
12
)
· [1|3.1.3.1] 5 eXd4 QXe2+ 6 KXe2 Ke6 = Bla ck King will se at o n d5 and White cannot
get through.
· [1|3.1.3.2] 5 eXf4 QXe2+ 6 KXe2 Ke6 = s ame as line above, the Black King seats on
d5 and Black may just move his Rook between b6 c6.
[1|3.2] 3 d4 e4 (
4 c4 17
, 4 f4 10
, 4 RXb4 9
, 4 Rb3 9
, 4 QXb5 8
, 4 Rc2
8
, 4 Qc4 8
)
4 cXb4 eXf3 (
5 Qd3 33
, 5 Rc2 9
, 5 QXb5 10
, 5 Rb3 9
, 5 Bc3 8
, 5 e4 7
,
5 Qc4 7
)
6
· [1|3.2.1] 5 QXf3 Rc6 (6 Rb3 31
, 6 e4 14
, 6 Bc3 12
, 6 Qf4 9
, 6 QXd5 8
,
6 Rc2 7
)
- [1|3.2.1.1]6 Qe2 Rc4 (
7 Rb3 29
, 7 Kf3 27
, 7 QXc4 17
)
* [1|3.2.1.1.1]7 Qf3 Qe4 (8 Ke2 32
, 8 Qe2 25
, 8 Rb3 21
)
8 QXe4 f Xe4 = White canno t get through since his Bishope is limited by his pawns. If
the White Rook moves to the third raw then . . . Rc2 limits the White choice to Rc3 after
the Rook exchange the position is an easy draw.
* [1|3.2.1.1.2] 7 Qd3 Qe4 = if White takes on e4 we have the same position as
variation [1|3.2.1.1.1] otherw ise Black just moves his king on e6-f6.
- [1|3.2.1.2] 6 Ke2 Rc4 = similarly to lines [1|3.2.1.1.1] and [1|3.2.1.1.2] Black will
play . . . Qe4 and block the position.
· [1|3.2.2] 5 KXf3 Qe4 6 Kf2 Ke6 (
7 Qc4 7
, 7 QXb5 8
, 7 Qd3 8
, 7 Rc2
12
, 7 Qf3 Rc6 = see line [1 |3.2.1.1.2]
)
7 Rb3 Rc6 (8 QXb5 34
) = if White does not take the b5 pawn the position is similar
to line [1|3.2.1.1.1].
2 cXb4 d4 (
3 Bc3 dXc3 8
, 3 Rb3 QXb311
)
[2|1] 3 eXd4 eXd4 (
4 Rb3 10
, 4 Ne3 29
, 4 Bc3 1 5
, 4 Be3 19
, 4 Bf4 11
)
[2|1.1] 4 f4 QXe2+ 5 KXe2 f4 = since only Kings ca n move without losing a piece and leading
to a lost position (Rb3 is poss ible but changes nothing to the evaluation of the position).
[2|1.2] 4 QXe6+ KXe6 5 f4 = similar as line [2|1.1].
[2|1.3] 4 NXd4 NXd4 (
5 Qe3 22
, 5 QXe6 16
, 5 Be3 10
)
5 Bc3
[2|2] 3 f4 eXf4 (
4 e4 24
, 4 Rb3 14
, 4 Bc3 1 1
, 4 Qf3 23
, 4 Kf3 15
)
[2|2.1] 4 eXd4 QXe2+ 5 KXe2 Ke6 = the Black King will move to d5-e6.
[2|2.2] 4 eXf4 QXe2+ 5 KXe2 = see variation [2|1.2].
[2|2.3] 4 NXd4 NXd4 5 eXd4 (5 Bc3 fXe3+ 6 QXe3 QXe3+ 7 KXe3 Be5 followed by exchanges
to a completly drawn endga me does not change the assesment of the position) 5. . . QXe2+ 6
KXe2 Ke6 = the Black King will seat on d5.
[2|4] 3 NXd4 NXd4
[2|4.1] 4 eXd4 eXd4 (
5 Rb3 9
, 5 Rc2 13
, 5 Bc3 11
, 5 Be3 19
, 5 Bf4 15
, 5 Qe3
8
, 5 Qe4 10
, 5 Qe5 7
)
[2|4.1.1] 5 f4 QXe2+ 6 KXe2 = see variation [1|3.1.1.1]
[2|4.1.2] 5 QXe6+ KXe6 = se e variation [1|3.1.1.2]
3 e4 f4 (
4 Rb3 QXb3 7
, 4 NXd4 NXd4 16
, 4 Ne3 f Xe3 18
, 4 Bc3 dXc3 9
, 4 Be3 18
,
4 Qe3 12
)
4 BXf4 eXf4 (5 e5 16
, 5 Rb3 7
, 5 NXd4 9
, 5 Ne3 7
, 5 Qe3 7
)
5 Qd2 Be5 = since the only non losing moves for White are limited to the King and Queen moves over
the d2, e2 and f2 squares.
4.2 White moves c4
1 c4 bXc4 The pin on the b file leads to forced mate 27
.
7
4.3 White moves d4
1 d4 e4 (
2 Nb4 21
, 2 QXb5 12
, 2 Qc4 9
, 2 Qd3 1 0
, 2 f Xe4 13
)
[1|1] 2 b4 c4 (
3 f Xe4 9
, 3 Rb3 8
, 3 QXc416
, 3 Qd3 9
) 3 f4 BXb4 (4 Rb3 8
, 4
RXb4 NXb4 21
, 4 QXc4 15
, 4 Qd3 10
, 4 Qf3 11
)
[1|1.1]4 cXb4 Qd6 = despite his extra piece White cannot win since he is blocked by his own
pawns on dark squares.
[1|1.2]4 NXb4 NXb4 = 5 RXb4 Qd6 and White may only move his Rook, on 5 bXc4 Qd6
is simila r to [1|1.1].
[1|2] 2 c4 bXc4 (3 b4 15
, 3 bXc4 19
, 3 Nb4 14
, 3 Bb4 10
, 3 f Xe4 10
, 3 f4 17
,
3 Qd3 8
, 3 QXc4 1 2
)
[1|2.1] 3 dXc5 BXc5 (
4 b4 20
, 4 Nb4 12
, 4 Nd4 15
, 4 Bb4 14
, 4 Qd3 8
, 4
QXc4 11
, 4 f4 12
)
4 Bc3 Qe5 (
5 BXe5+ 19
, 5 bXc4 16
, 5 f Xe4 13
, 5 b4 9
, 5 Nd4 12
, 5 Nb4
12
, 5 f4 7
, 5 Bd4 9
, 5 QXc4 9
, 5 Qc3 7
)
5 Qd2 f4 (
6 Ke2 12
, 6 f Xe4 13
, 6 b4 17
, 6 Nd4 10
, 6 Bd4 8
, 6 bXc4 8
,
6 Nb4 8
, 6 Qd4 7
, 6 Bb4 7
, 6 Qe2 6
, 6 QXd5 6
, 6 Qd3 5
)
6 BXe5 NXe5 (
7 Nd4 16
, 7 Qc312
, 7 f Xe4 20
, 7 Qd4 11
, 7 Ke2 11
, 7
Nb4 10
, 7 QXd5 8
, 7 Qb4 6
, 7 bXc4 5
, 7 Qe2 5
, 7 Qd3 4
)
7 b4 f Xe3+ (
8 QXe3 14
, 8 Ke2 1
)
8 NXe3 Nd3+ (
9 QXd3 6
)
9 Ke2 Nf4+ 10 Kf2 Nf4+ = draw by repetition.
[1|2.2] 3 Bc3 RXb3
4 RXb3 cXb3 (
5 f Xe4 9
, 5 f4 15
, 5 Nb4 11
, 5 Bb4 8
, 5 Bb2 10
, 5 Bd2 6
, 5
Qb5 6
, 5 Qc4 6
, 5 Qd3 5
, 5 Qd2 15
)
5 dXc5+ Be5 (
6 f Xe4 8
, 6 f4 6
, 6 Nb4 7
, 6 Bb2 8
, 6 Bb4 9
, 6 Bd2 7
, 6
Bd4 9
, 6 Qd2 16
, 6 Qd3 4
, 6 Qc4 6
, 6 Qb5 6
, 6 BXe5+ 13
)
6 Nd4 BXd4 (
7 f Xe4 7
, 7 f4 6
, 7 Bb2 11
, 7 Bb4 5
, 7 Bd2 6
, 7 Qd2 11
,
7 Qd3 3
, 7 Qc4 8
, 7 Qb5 10
, 7 Qb2 11
, 7 Qc2 2
)
- [1|2.2.1] 7 BXd4+ NXd4 (
8 Qd211
, 8 f Xe46
, 8 c6Q5
(other promotions as well), 8
f43
, 8 Qc2 bXc2Q+ checkmate )
8 Qb2 f4 (
9 QXd4+17
, 9 c6Q11
(other promotions a s well), 9 f Xe47
, 9 Qd26
, 9
Qc2 bXc2Q+ checkmate )
9 eXd4 eXf3 (
10 Qd23
, 10 c6B Qe3 checkmate, 10 c6N Qe3 checkmate, 10 Qc2 Qe3
checkmate, 10 Qe2 f Xe2Q+ checkmate, 10 QXb3 Qe2 checkmate, 10 Qc3 Qe2 checkmate
)
10 c6Q QXc6, promotion to Rook is handled similarly, (
11 KXf37
, 11 Qc34
, 11 Qd24
,
11 Qc23
, 11 Qe23
)
11 QXb3 Qe6 = Black will play . . . Qe2+ and after Queen exchange the pawn e ndgame is draw.
- [1|2.2.2] 7 eXd4 e3+ 8 QXe3 QXe3+ 9 KXe3 = the Black King just moves to e6-f6 and
White King cannot break through. If the White Biwhop goes to e5 either Black can play f4 and
get room for his King or it means that White played f4 hence after . . . Nb4 the Knight cannot be
taken without stalemating the Black King.
[1|3] 2 dXc5 BXc5 (3 c4 b42 3
, 3 f Xe4 14
, 3 f4 Qd614
, 3 Nb4 f416
, 3 QXb5 11
,
3 Qc4 9
, 3 Qd3 9
, 3 b4 16
)
3 Nd4 NXd4 (
4 b4 12
, 4 c4 9
, 4 f Xe4 10
, 4 f4 9
, 4 Rc2 9
, 4 QXb5 8
, 4 Qc4
8
, 4 Qd3 7
)
8
[1|3.1] 4 cXd4 eXf3 (5 b4 12
, 5 dXc5 14
, 5 e4 5
, 5 Rc2 10
, 5 Bc3 10
, 5
Bb4 8
, 5 Qc4 12
, 5 QXb5 16
)
· [1|3.1.1] 5 Qd3 Bd6 = on any reasonable move (
6 Qe4 6
, 6 Qc2 3 2
, 6 b4 34
,
6 Bc3 24
, 6 Bb4 15
, 6 Qc3 24
, 6 e4 12
, 6 QXb5 8
, 6 Qe2 8
, 6 Qc4
7
, 6 QXf5+ 6
) Black plays . . . Qe4 and locks the position as in variation [1|3.1.3.1].
· [1|3.1.2] 5 QXf3 Bd6 = on any reasonable move (
6 QXf5+ 7
, 6 Qf4 8
, 6 QXd5
9
, 6 Qe4 4
6 e4 37
, 6 Bb4 22
) Black plays . . . Qe4 and locks the position as
in variation [1|3 .2.3.1].
· [1|3.1.3] 5 KXf3 Qe4+ 6 Kf2 Bd6 (
7 Rc2 10
, 7 Bb4 12
, 7 Qd3 7
, 7 Qc4
6
, 7 QXb5 8
)
[1|3.1.3.1]7 b4 Ke6 = Black just moves his King on e6 -f6 and the position is blo cked
on the da rk squares 8 Qf2 Kf6 9 QXe3 f Xe3
[1|3.1.3.2]7 Bc3 Ke6 = see line [1|3.2.3.1].
[1|3.1.3.3]7 Qf3 Ke6 = see line [1|3.2.3.1].
[1|3.2] 4 eXd4 Bd6 (
5 Rc2 f4 23
, 5 b4 f4 18
, 5 Be3 f4 22
, 5 Qe3 f4 22
, 5 c4
bXc4 23
, 5 Ke3 f4+ 23
, 5 Bf4 11
, 5 QXb5 10
, 5 Qc4 8
, 5 Qd3 4
, 5 Bf4
11
)
- [1|3.2.1 ] 5 f Xe4 QXe4 = if White exchanges Queen on e4 then with . . . fXe4 B lack closes
the position and with . . . Rc6 White cannot do anything. If White does not exchange Queens
then Black may just play his King (on 6 b4 f4 is 28
).
- [1|3.2.2] 5 f4 e3+ = since Black follows with . . . Qe4 and blocks the position.
[1|4] 2 f4 c4 (
3 Qd3 12
, 3 QXc4 bXc4 15
, 3 Qf3 8
)
[1|4.1] 3 b4 BXb4 = due to the blocked position White cannot achieve anything, this type
of positio n has already been treated in line [1|1] of this oracle fo r instance.
[1|4.2] 3 bXc4 dXc4 (
4 d5 QXd5 22
, 4 RXb5 RXb5 10
, 4 Rb3 9
, 4 QXc4 8
,
4 Qd3 9
, 4 Qf3 11
, 4 NXb4 Bc5 12
this surprising move lead to direct checkmate
since White is completly blo cked and will eventually, due to his lack of space, have to gite his
Queen within a few moves.
)
4 Rb4 NXb4 (
5 d5 8
, 5 NXb4 23
, 5 QXc4 5
, 5 Qd3 7
, 5 Qf3 6
) 5 cXb4
Qd5 = because the po sition is totally blocked and Black just moves his King to e6 f6. The
only way to untangle for White is to sacrifice the Queen on c4 which lead to quick checkmate.
[1|4.3] 3 Nb4 NXb4 (
4 Rc2 8
, 4 QXc4 9
, 4 Qd3 7
, 4 bXc4 15
, 4 Qf3
7
) = The draw is tricky to unders tand at first sight but becomes clear with the following
variation 4 cXb4 BXf4 ( 5 Bc3 16
, 5 Rc2 15
, 5 Qd3 6
, 5 QXc4 9
, 5 Qf3
7
). From here the idea is to build a blockade on dark squa res.
· After 5 eXf4 Rc6 (in order to be able to take with the Rook in the case of bXc4) = The
blockade has been achieved and Black just moves his Queen on d6 and his King on e6 f6.
· 5 bXc4 bXc4 6 eXf4 Rb5 = another blockade is built on dar k squares and White cannot
break through.
4.4 White moves e4
1 e4 f4 (
2 d4 25
, 2 Nb4 17
, 2 Nd4 25
, 2 BXf4 eXf4 22
, 2 Qe3 f Xe3+ 25
)
[1|1] 2 b4 cXb4 (
3 c4 15
, 3 d4 27
, 3 eXd5 27
, 3 Rb3 17
, 3 RXb4 34
, 3 NXb4
19
, 3 Nd4 14
, 3 Ne3 17
, 3 Be3 14
, 3 BXf4 16
, 3 Qe3 17
)
3 cXb4 d4 = This po sition is draw for the same reason as position [1|1.2] of the White oracle
(see section 3). White is in zugzwang and must give a piece, the only non losing way to do it is
by 4 BXf4 eXf4 5 Qd2.
9
[1|2] 2 c4 bXc4 (3 b4 22
, 3 dXc4 19
, 3 d4 15
, 3 Nb4 12
, 3 Nd4 15
, 3 Ne3
16
, 3 Bc3 2 0
, 3 Bb4 12
, 3 Be3 13
, 3 BXf4 12
, 3 Qe3 14
)
[1|2.1] 3 bXc4 RXb2 = is a tricky draw in which White appear to be losing but can hold.
The mainline is the following 4 cXd5 QXd5 5 eXd5 Nd4 6 NXd4 cXd4 at this point Black
will regain the Queen and the bishop by force (otherwise White g et mated) and end up in a
ending like this one 7 Bb4 RXe2+ 8 KXe2 BXb4 and the pos ition is a curious draw (clearly
White cannot win which is enough for our oracle).
[1|2.2] 3 cXd5 QXd5 (
4 b4 11
, 4 d4 10
, 4 Nb4 9
, 4 Nd4 8
, 4 Ne3 11
, 4
Bb4 7
, 4 Bc3 11
, 4 Be3 7
, 4 BXf4 9
, 4 Qe3 11
, 4 Qe4 11
, 4 QXe5+
3
)
· [1|2.2.1]4 bXc4 RXb2 = see line [1|2.1].
· [1|2.2.2]4 dXc4 Qe6 + since White is restricted by Black pawns that completly control
the dark squares and cannot move his Knight, hence his Rook. Black may just move his
Queen between e6 f5. (
5 b4 cXb4 22
).
[1|3] 2 Ne3 f Xe3+ (
3 QXe3 25
, 3 KXe3 22
)
3 BXe3 d4 + White cannot win 4 Bc2 dXc3 5 BXc3 b4 6 Bd2 Nd4 7 Qe3 Rc6 White
is in zugzwang and must give another piece.
[1|4] 2 Be3 d4 (
3 Nb4 f Xe3+ 9
, 3 NXd4 f Xe3+ 17
, 3 Bd2 25
, 3 BXd4 cXd413
,
3 BXf4 dXc3 16
, 3 Qd2 28
3 b4 dXc3 11
Black either promotes c pawn or is a Rook
up (and 4 Rb2 QXb2 5 BXc5 Nd4 is not helping)
)
[1|4.1] 3 c4 f Xe3+ (
4 QXe3 13
)
4 NXe3 dXe3+ = b e cause on each recapture by White Black closes the position with . . . b4
and white cannot break through since f4 leads to a quick defeat.
[1|4.2] 3 cXd4 cXd4 = (
4 b4 17
, 4 Nb4 9
, 4 NXd4 14
, 4 Bd2 14
, 4 BXd4
NXd4 12
, 4 Qd2 15
) 4 BXf4 eXf4 5 b4 (otherwise White is a piece down and will
lose) 5. . . Be5 and the position is completely blocked on the dark square s.
2 eXd5 QXd5 (
3 d4 eXd4 21
, 3 Nb4 cXb4 11
, 3 Nd4 eXd4 16
, 3 Ne3 f Xe3+ 52
, 3
Be3 f Xe3+ 44
, 3 BXf4 eXf4 15
, 3 Qe3 f Xe3 10
, 3 QXe5+ NXe5 2
)
[2|1] 3 b4 cXb4 (
4 c4 16
, 4 d4 15
, 4 Rb3 12
, 4 RXb4 14
, 4 NXb4 13
, 4 Nd4
12
, 4 Ne3 14
, 4 Be3 10
, 4 BXf4 10
, 4 Qe3 9
, 4 Qe4 14
, 4 QXe5+ 5
)
4 cXb4 Nd4 (
5 Rb37
, 5 NXd48
, 5 Ne314
, 5 Bc39
, 5 Be39
, 5 Qe32
, 5
QXe5+5
, 5 BXf48
)
5 Qe4 QXe4 = (
6 Rb3 Qe2 checkmate, 6 NXd4 5
, 6 Ne3 QXf3 checkmate, 6 Bc3 QXf3
checkmate, 6 Be3 QXf3 checkma te, 6 BXf4 QXf3 checkmate
)
on either d-pawn or f-pawn capture of the Queen Black plays . . . NXc2 and then his King on e6-f6
squares. The position is completly blocked.
[2|2] 3 c4 bXc4 by transposition we have reached line [1|2.2].
3 Qe4 QXe4 (4 b4 7
, 4 c4 6
, 4 d4 7
, 4 Nb4 10
, 4 Nd4 6
, 4 Ne3 7
, 4 Be3 QXe3
checkmate, 4 BXf4 8
)
[3|1] 4 dXe4 b4 (
5 NXb427
, 5 Nd49
, 5 BXf47
, 5 Be36
)
[3|1.1] 5 c4 Ke6 = the position is completely blocked and Black can just move his King on
e6-f6.
[3|1.2] 5 cXb4 cXb4 = White pieces are blocked, his only active plan is to bring the King
on c4 but Black can play its bishop on c5-f2 6 Ke2 Bc5 7 Kd3 Ke6 8 Kc4 Kd6.
10
[3|1.3] 5 Ne3 f Xe3+ (6 BXe3 10
, 6 Ke2 14
)
6 KXe3 bXc3 ( 7 b4 5
, 7 Rc2 11
, 7 Kd3 5
, 7 Ke2 5
, 7 Kf2 5
, 7 f4 5
)
7 BXc3 c4 (
8 b423
, 8 Ke215
, 8 Kd228
, 8 Kf213
, 8 f425
, 8 Bb49
, 8
Bd47
, 8 BXe5+8
, 8 Bd29
, 8 Rc210
, 8 Rd210
, 8 Re28
, 8 Rf28
)
8 bXc4 RXb2 + 9 BXb2 Ke6 white cannot win since after . . . Bc5 White clearly cannot
progre ss.
[3|1.4] 5 Ke2 bXc3 ( 6 Kd3 5
, 6 BXf4 5
, 6 Nb4 5
, 6 Be3 5
, 6 b4 5
, 6
Ne3 5
, 6 Nd4 3
, 6 Kf2 cXd2Q+ checkmate)
6 BXc3 Nd4+ (7 Kd2 39
, 7 Kf2 16
, 7 NXd4 21
)
· [3|1.4.1]7 Kd3 NXf3 (
8 Kc4 40
, 8 Ke2 21
, 8 Bd2 14
, 8 Nb4 11
, 8 Bb4
13
, 8 Nd4 13
, 8 BXe5 10
, 8 Bd4 10
)
- [3|1.4.1.1] 8 b4 Rc6 ( 9 Ke2 41
, 9 Ke2 24
, 9 Bd2 14
, 9 Kc4 12
,
9 Bd4 12
, 9 Nd4 11
, 9 BXe5 10
, 9 Rb3 9
)
* [3|1.4.1.1.1] 9 b5 Rb6 =
*[3|1.4.1.1.2] 9 bXc5 BXc5 =
* [3|1.4.1.1].3 9 Ne3 f Xe3 =
-[3|1.4.1.2] 8 Ne3 f Xe3 (
9 Ke2 41
, 9 Re2 24
, 9 b4 17
, 9 Rc2 15
,
9 Kc4 14
, 9 Bb4 13
, 9 BXe5 8
, 9 Rd2 7
, 9 Bd4 7
, 9 Bd2 7
, 9
Rf2 5
, 9 Kc2 2
)
9 KXe3 c4 = since Rook exchanges is unavoidable (otherwise White lose) and the Bishop’s
ending is draw.
· [3|1.4.2] 7 BXd4 eXd4 ( 8 NXd4 18
, 8 Kd2 18
, 8 e5 13
, 8 Nb4 13
, 8
Ne3 10
, 8 Kf2 8
, ) = 8.b4 c4 9.Nxd4 Be5 and the resulting Rook ending is
draw.
4 f Xe4 b4 (
5 d4 14
, 5 NXb4 21
, 5 Nd4 13
, 5 Be3 7
, 5 BXf4 7
)
[4|1] 5 Ke2 Ke6 6 Kf2 (or 6 Kf3) 6. . . Kf6 = since other moves than King loses (see previous
lines) or rejoin the mainline (c4).
[4|2] 5 Kf3 Ke6 same as line [4|1].
5 c4 Nd4 = the position is totally locked on dark squares and White cannot progress.
4.5 White moves f4
1 f4 eXf4 (
2 Nb4 20
, 2 Nd4 24
, 2 Qf3 24
, 2 Kf3 Ne5+ 17
, 2 c4 Be5 26
, 2 e4 f Xe4
18
)
[1|1] 2 b4 f3 (
3 bXc5 14
, 3 c4 11
, 3 d4 11
, 3 e4 9
, 3 Rb3 9
, 3 Nd4 11
, 3
KXf3 24
)
3 QXf3 Ne5 (
4 c4 14
, 4 d4 14
, 4 e4 11
4 Rb3 13
, 4 Nd4 19
, 4 Qe2 23
, 4
Qf4 9
, 4 QXf5+ 2
, 4 Qe4 7
, 4 QXd5 6
, 4 Ke2 20
)
4 bXc5 NXd3+ 5 Ke2 NXc5 (
6 Nb4 37
6 c4 1 4
, 6 e4 11
, 6 Rb3 12
, 6 RXb5 14
,
6 Qf2 21
, 6 Qf4 12
, 6 Qe4 8
, 6 QXd5 10
, 6 QXf5+ 9
)
[1|1.1] 6 Rb4 Ne4 (
7 c4 12
, 7 Rb3 21
, 7 Rb2 37
, 7 Kd3 29
, 7 Qf2 13
, 7
Qf4 15
)
7 Nd4 NXd2 8 KXd2 Qe4 = 9 QXe4 f Xe4 10 RXb5 RXb5 11 NXb5 and the ending
Knight vs. Bishop is draw or 9 RXb5 QXf3 10 RXb6 Qf2+ 11 Kd3 Ke6 12 Nc6+
and perpetual check.
11
[1|1.2] 6 Nd4 Qe4 Because of the threat to the White king, White’s move is forced 7 QXe4
f Xe4 + White is blocked and must take the b pawn with his Rook if he looks for any
progre ss after the rook exchanges Black easily draw. If 8 NXb4 Nc2 9 Rb3 Bc5 White is
in zugzwang and lose.
[1|3] 2 d4 f Xe3+ (
3 NXe3 cXd4 26
, 3 QXe3 41
, 3 Kf3 Qe4 checkma te)
3 BXe3 Qe4 (4 Qf3 48
, 4 Qd2 30
, 4 b4 35
, 4 c4 16
, 4 Nb4 15
, 4 Bd2 24
,
4 Bf4 18
, 4 Qd3 9
, 4 Qc4 8
, 4 QXb5 8
)
4 dXc5 BXc5 (5 c4 4
, 5 Nb4 14
, 5 Nd4 14
5 Qd2 5
, 5 Qd3 5
, 5 Qc4 5
, 5
QXb5 5
, 5 Qf3 32
, 5 Bc4 1 5
)
[1|3.1] 5 b4 Bd6 the o nly non losing move is 6 BXb6 Qf4+ = perpertual check on d2 f4.
[1|3.2] 5 BXc5 Qf4+ = perpertual check on d2 f4.
2 eXf4 QXe2+ 3 KXe2 d4 (
4 Nb4 cXb4 24
, 4 NXd4 cXd4 31
, 4 Ne3 dXe3 31
, 4 Be3
dXe3 8
)
[4|1] 4 b4 Ke6 (5 Ne3 32
, 5 Be3 40
, 5 NXd4 28
)
[4|1.1] 5 bXc5 BXc5 (6 Kf3 23
, 6 Rb4 23
, 6 Ne3 22
, 6 RXb5 12
, 6 Kf2 7
)
· [4|1.1.1] 6 c4 bXc4 (
7 Rb3 6
, 7 Rb4 10
, 7 Rb5 8
, 7 Nb4 9
, 7 NXd4 6
,
7 Ne3 10
, 7 dXc4 8
, 7 Bc3 12
, 7 Bb4 10
, 7 Be3 7
, 7 Kf2 6
, 7 Kf3
7
)
7 RXb6 BXb6 (8 Nb4 11
, 8 NXd4 13
, 8 Ne3 14
, 8 Bb4 15
, 8 Bc3
14
, 8 Be3 14
, 8 Kf2 10
, 8 Kf3 10
)
8 dXc4 Bc5 + White is blocked and cannot untangle if B lack just moves his King to
d6-e6.
· [4|1.1.2] 6 cXd4 NXd4 (
7 Kf213
, 7 Ke313
) 7 NXd4+ BXd4 = the Black King
blocks the positio n on d5.
· [4|1.1.3] 6 Rb3 dXc3 (7 RXb5 11
, 7 d4 15
, 7 Rb4 11
, 7 Rb2 5
, 7 Nb4 4
,
7 Nd4+ 4
, 7 Ne3 2
, 7 Kf3 4
)
- [4|1.1.3.1] 7 BXc3 b4 Black moves his King to d5 and blocks the position.
- [4|1.1.3.2] 7 RXc3 Nd4+ (8 Ke3 11
)
* 8 NXd4+ BXd4 = Black King comes to d5 and blocks the position.
* 8 Kf2 Kd5 =
- [4|1 .1.3.3] 7 Be3 BXe3 = after . . . b4 and . . . Kd5 Black lo cks down the positio n
and White cannot progress.
· [4|1.1.4] 6 Nb4 dXc3 (
7 Rc2 8
, 7 NXc6 7
, 7 d4 6
, 7 Kf3 5
, 7 Rb3 4
, 7
Nd5 5
, 7 Nc25
)
7 BXc3 NXb4 (8 RXb4 17
, 8 Ke2 17
, 8 Rd2 16
, 8 Be5 16
, 8 Rb3 15
,
8 Bd2 14
, 8 Kf3 14
, 8 Bf6 13
, 8 Bd4 11
, 8 Rc3 9
)
[4|1.1.4.1] 8 BXb4 BXb4 = this Rook ending is clearly draw.
[4|1.1.4.2] 8 d4 Bd6 (
9 d5+ 18
, 9 Rb3 16
, 9 RXb4 15
, 9 Rc2 10
, 9
Rd2 14
, 9 Bd2 18
, 9 Kd2 18
, 9 Ke3 16
, 9 Kf3 19
, 9 Kf2 12
)
9 BXb4 BXb4 = this is the same ending as variation [1|1.1.4.1].
· [4|1.1.5] 6 NXd4+ NXd4 (7 Ke3 23
, 7 Kf2 25
)
7 cXd4 BXd4 (8 Rc2 20
, 8 RXb5 10
, 8 Bc3 11
, 8 Bb4 11
, 8 Be3 21
, 8
Kf3 10
) = since Black King may move to seat on d5 and block the position. If Bishops
are exchanged the resulting Rook ending is clearly draw.
· [4|1.1.6] 6 Be3 dXc3 (7 d4 9
, 7 Rb4 10
, 7 RXb5 12
, 7 Nb4 5
, 7 Nd4+
9
, 7 Kf2 7
, 7 Kf3 7
, 7 Bd2 5
, 7 Bf2 7
, 7 Bd4 9
, 7 BXc5 7
)
7 Rb3 BXe3 (
8 d4 11
, 8 Rb2 5
, 8 Rb4 9
, 8 RXb5 7
, 8 Nb4 10
, 8
Nd4+ 5
, 8 NXe3 8
, 8 Kf3 9
, 8 RXc3 13
)
12
8 KXe3 b4 (9 RXb4 7
, 9 RXc3 9
, 9 Rb2 5
, 9 NXb4 11
, 9 Nd4 6
, 9
Ke2 15
, 9 Kf2 12
, 9 Kf3 18
)
9 d4 Kd5 (
10 RXb4 5
, 10 RXc3 8
, 10 Rb2 4
, 10 NXb4+ 14
, 10 Ke2
9
, 10 Kf2 9
, 10 Kf3 8
)
10 Kd3 Rb5 (11 RXb4 8
, 11 RXc3 17
, 11 Rb2 6
, 11 Ke2 6
, 11 NXb4+
15
, 11 Ke3 8
)
11 Ne3+ Kd6 (12 Rb2 8
, 12 RXb4 10
, 12 RXc3 16
, 12 Kc2 10
, 12
Ke2 9
, 12 Kc4 16
, 12 Nc4+ 19
, 12 Nd5 7
, 12 NXf5+ 9
, 12 d5
14
)
12 Nc2 Kd5 = by r epetition.
[4|1.2] 5 c4 bXc4 (
6 bXc6 10
, 6 b5 19
, 6 Rb3 7
, 6 NXd4 ♯♯
, 6 Ne3 12
, 6 Bc3
8
, 6 Be3 11
, 6 Kf2 12
, 6 Kf3 11
)
6 dXc4 cXb4 (7 c5 20
, 7 RXb4 12
, 7 NXb4 24
, 7 NXd4+ 22
, 7 Bc3 12
, 7
BXb4 20
, 7 Be3 16
, 7 Kf3 21
, 7 Kf2 22
)
· [4|1.2.1] 7 Ne3 dXe3 (
8 Rb3 8
, 8 RXb4 7
, 8 Rc2 10
, 8 Bc3 7
, 8 BXb4
10
, 8 Kd3 11
, 8 Kf3 11
, 8 KXe3 ♯♯
, 8 c5 11
)
8 BXe3 Ne5 (9 Rb3 19
, 9 RXb4 13
, 9 Rc2 12
, 9 Rd2 23
, 9 Bd2 13
,
9 Bf2 20
, 9 Bd4 15
, 9 Bc5 12
, 9 BXb6 19
, 9 Kd2 12
, 9 Kf2 12
)
[4|1.2.1.1] 9 c5 BXc5 = 10 BXc5 Nc4 11 RXb4 RXb4 12 BXb4.
[4|1.2.1.1] 9 f Xe5 BXe5 = 10.c5 Rc6 1 1.Rxb4 Kd5
· [4|1.2.2] 7 Kd3 Bc5 (
8 RXb4 24
, 8 NXb4 24
, 8 NXd4+ 24
, 8 Ne3 21
, 8
BXb4 24
, 8 Bc3 1 1
, 8 Be3 21
, 8 Ke2 18
)
8 Rb3 Kd6 9 Ke2 Ke6 = White is blocked and cannot do anything concrete in this
position.
· [4|1.2.3] 7 Rb3 Bc5 = this move simply transposes to variation [4|1.2 .2]
[4|1.3] 5 cXd4 cXd4 = the position is totally blocked Black just moves his King on e6 d5.
[4|1.4] 5 Rb3 c4 (
6 dXc4 24
, 6 cXd4 13
, 6 NXd4 10
)
6 Rb2 dXc3 (7 dXc4 10
, 7 Ke3 6
, 7 Be3 6
)
7 BXc3 cXd3+ (
8 Kf3 15
, 8 Ke3 19
, 8 Kd2 13
)
8 KXd3 BXf4 (9 Rb3 33
, 9 Bd4 29
, 9 Ke2 26
, 9 Be5 9
, 9 Bf6 18
, 9 Bd2
22
)
9 Nd4+ NXd4 (
10 Rb3 9
, 10 Rc2 10
, 10 Rd2 11
, 10 Re2+ 9
, 10 Rf2
17
, 10 Bd2 14
, 10 KXd4 1 1
)
10 BXd4 Rc6 = Black will place his King on d5 and White cannot progress.
[4|1.5] 5 Kf2 dXc3 (
6 bXc5 4
, 6 d4 5
, 6 Rb3 4
, 6 Nd4 10
, 6 Ne3 10
, 6 Be3
7
, 6 Ke2 6
, 6 Ke3 6
, 6 Kf3 5
)
6 BXc3 cXb4 (7 d4 13
, 7 Rb3 16
, 7 RXb4 23
, 7 Nd4+ 24
, 7 Ne3 11
, 7 Bf6
18
, 7 Be5 13
, 7 Ke2 12
, 7 Ke3 13
, 7 Kf3 13
)
· [4|1.5.1] 7 BXb4 NXb4 (
8 d4 1 1
, 8 Rb3 10
, 8 RXb4 18
, 8 Nd4 13
, 8 Ne3
13
, 8 Ke2 12
, 8 Ke3 13
, 8 Kf3 12
)
8 NXb4 BXb4 (9 d4 11
, 9 Rb3 12
, 9 Rc2 11
, 9 Rd2 7
, 9 Re2+ 12
, 9
Ke2 11
, 9 Ke3 10
, 9 Kf3 11
) 9 RXb4 Kd5 = this Rook endg ame is a draw.
· [4|1.5.2] 7 Bd2 b3 (
8 Ne3 33
, 8 Nb4 14
, 8 Be3 23
, 8 d4 14
, 8 Bc3
14
, 8 Ke3 21
, 8 Bb4 11
, 8 Nd4 12
, 8 Kf3 12
, 8 Ke2 12
)
8 RXb3 Bc5+ (
9 Ne3 23
, 9 Nd4+ 12
)
* [4|1.5.2.1] 9 Be3 BXe3+ = the Black King will block the positio n on d5.
* [4|1.5.2.2] 9 d4 NXd4 = 10 NXd4+ NXd4 11 Rd3 b4 12 Be3 b3 13 BXd4
the White B ishop must be exchanged vs the b pawn and the Rook ending is draw.
* [4|1.5.2.3] 9 Ke2 Nd4+ = 10 NXd4+ BXd4 and Black locks the position by
. . . Kd5.
13
* [4|1.5.2.4] 9 Kf3 Kd5 = the reason has to be seen in previous lines. Black exchange
the Bishop vs the Knight and the remaining position is blocked.
· [4|1.5.3] 7 Bd4 NXd4 (
8 Rb3 8
, 8 RXb4 10
, 8 NXb4 10
, 8 Ne3 9
, 8 Ke3
8
)
8 NXd4+ Kd5 (
9 Rb3 9
, 9 RXb4 10
, 9 Nb3 31
, 9 NXb5 10
, 9 Nc6 11
,
9 Ne2 22
, 9 Ne6 12
, 9 Nf3 23
, 9 Rc2 11
, 9 Rd2 12
, 9 Re2 12
, 9 Ke2
11
, 9 Ke3 11
, 9 Kf3 11
)
- [4|1.5.3.1] 9 Nc2 Bc5+ = for the same reasons as lines [4|1.5.2.1] / [4|1.5.2.2] / [4|1.5.2.3]
/ [4|1.5.2.4]
- [4|1.5.3.2] 9 NXf5 Bc5+ = once again the Black Bishop is exchanged vs the Knight
and the remaining Rook e nding is dr aw.
[4|1.5.4] 7 NXb4 BXb4 all moves but one lead to White checkmate (
8 d4 10
, 8 Rb3
17
, 8 RXb4 11
, 8 Rc2 11
, 8 Rd2 10
, 8 Re2+ 13
, 8 Bd2 11
, 8 Bd4 7
,
8 Be5 11
, 8 Bf6 12
, 8 Ke2 9
, 8 Ke3 10
, 8 Kf3 9
) 8 BXb4 NXb4 = (9 d4
8
, 9 Rb3 11
, 9 Rc2 7
, 9 Rd2 12
, 9 Re2+ 12
, 9 Ke3 12
, 9 Ke2 12
, 9
Kf3 11
)
9 RXb4 leads to the same drawn Rook endgame as [4|1.5.1].
[4|1.6] 5 Kf3 cXb4 (
6 Kf2 13
, 6 c4 27
, 6 RXb4 17
, 6 Rb3 14
, 6 Be3 13
, 6
Ke2 14
, 6 Ne3 11
)
[4|1.6.1] 6 cXd4 b3 (
7 Kf2 15
, 7 Ke2 11
, 7 Ke3 14
, 7 Bc3 19
, 7 Bb4 11
,
7 Be3 26
, 7 Ne3 15
, 7 Nb4 14
)
[4|1.6.1.1]7 RXb3 b4 (8 BXb435
, 8 Rc3 17
, 8 RXb4 20
, 8 NXb4 15
,
8 Ne311
) = White cannot progress wihtout giving a piece or moving d5+ after which
the Black King blocks the position. If White moves around Black simply plays his Rook on
b5-b6. 8 d5+ KXd5 9 Ne3+ Ke6 (
10 Ke211
, 10 Nd5 11
, 10 Rc3 14
, 10
Bc3 33
, 10 Rb2 33
, 10 RXb4 20
, 10 BXb4 17
, 10 NXf5 24
, 10 Nc4 14
,
10 Kf2 29
, 10 d4 11
) and the only non losing line is to repeat with 10 Nc2.
[4|1.6.1.2]7 d5+ = see line [4|1.6 .1.1].
[4|1.6.2] 6 cXb4 Kd5 = position is totally blocked.
[4|1.6.3] 6 NXd4+ NXd4 = 7 cXd4 Black puts his King on d5 and the White position is
totally blocked.
[4|1.6.4] 6 NXb4 NXb4 = 7 cXb4 Kd5 is similar to line [4|1 .6.2].
[4|2] 4 cXd4 cXd4 = see line [4|3].
[4|3] 4 Kf2 b4 = the position is totally blocked and Black can just move his King on e6-f6 5 cXb4
cXb4 6 Kf3 Ke6 etc.
[4|4] 4 Kf3 b4 = for the same reas ons as in line [4|3].
4 c4 b4 = see line [1|5.2]
4.6 White moves Nb4
1 Nb4 cXb4 21
White is a piece down.
4.7 White moves Nd4
1 Nd4 eXd4 25
White is a piece down.
14
5 Conclusion
The game-theoretical value of Gardner’s chess has be en proved to be a draw. The proof was done in a
semi-automated way in which humans were guiding the engine. The authors were ’pushing’ lines for which
it was thought that the exact distance to checkmate could be computed and backtracked once leaves were
showing perfect dista nc e to checkmate. This meta-algorithm leads to a very asymmetric way of selecting
moves. For instance, when a position is thought to be decidable as a White win, very few time is spent on
White decision nodes (since we ’know’ the game to be won more or less no matter what). The ide a is tha t
enormous time and energy can be saved w he n the game theoretic value of a position, rather than the mo st
precise move or the shortest path to checkmate, is looked for. Indeed, when a game is thought to be winning ,
e.g. for White, one has only to provide one for ced line (even if it is not the ’best’ one) and thus can avoid
exhaustive search at White decision nodes. It can be seen as a fo rm of meta-negascout [Fis81]. Nevertheless
it is very different in the sense that the process is very asy mmetric and guided by the fact that the overall
evaluation of the position is known.
This procedure can be fully auto mated and tuned to some given degree of precision (basically what is
the threshold after which a position is co nsidered as decided). For future works we plan to implement it and
test it for larger chess variants in order to compute their game theoretic values. Other games could also be
considered.
Acknowledgments
We thank Fra nc ois C hallier and Philippe Virouleau for their technical and technological help.
References
[All94] V. Allis. Searching for Solutions in Games and Artificial Intelligence. PhD thesis, University of
Limbur g, Maastricht, The Netherlands, 1994.
[Fis81] J. P. Fishburn. Analysis of Speedup in Distributed Algorithms. PhD thesis, University of Win-
consin, Madiso n, 1981.
[Kry04] K. Kryukov. 3 × 3 chess. website : http://kirr.homeunix.org/chess/3x3-chess/, 2004.
[Kry09] K. Kryukov. 3 × 4 chess. website : http://kirr.homeunix.org/chess/3x4-chess/, 2009.
[Kry11] K. Kryukov. 4 × 4 chess. website : http://kirr.homeunix.org/chess/4x4-chess/, 2011.
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[Pri07] D. B. Pritchard. The Classified Encyclopedia of Chess Variants. J. Beasley, 2007.
[Pro12] F. Prost. On the impact of information technologies on society: an historical perspective through
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EasyChair, 2012.
[RCK
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10] T. Romstad, M. Costalba, J. Kiiski, D. Yang, S. Spitaleri, and J. Ablett. Stockfish. web site:
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[SBB
+
07] J. Schaeffer, N. Burch, Y. Bjornsson, A. Kishimoto, M. Muller, R. Lake, P. Lu, and S. Sutphen.
Checkers is solved. Nature, 317(5844):1518–1522, 2007.
15
... Example. An exemplary game called Gardner 1 , formatted according to Simplified Boardgames grammar is presented partially in Figure 2. It is 5 × 5 chess variant proposed by Martin Gardner in 1969 and weakly solved in 2013 [6] -the game-theoretic value has been proved to be a draw. ...
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Full-text available
We formalize Simplified Boardgames language, which describes a subclass of arbitrary board games. The language structure is based on the regular expressions, which makes the rules easily machine-processable while keeping the rules concise and fairly human-readable.
... Regular Gardner (recently weakly solved [23]) is one of the minichess variants, played on a 5 × 5 board with a 100 turn limit. The starting position looks as in the regular chess with removed columns f , g, h, and rows 3, 4, 5. ...
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The game of checkers has roughly 500 billion billion possible positions (5 × 1020). The task of solving the game, determining the final result in a game with no mistakes made by either player, is daunting. Since 1989, almost continuously, dozens of computers have been working on solving checkers, applying state-of-the-art artificial intelligence techniques to the proving process. This paper announces that checkers is now solved: Perfect play by both sides leads to a draw. This is the most challenging popular game to be solved to date, roughly one million times as complex as Connect Four. Artificial intelligence technology has been used to generate strong heuristic-based game-playing programs, such as Deep Blue for chess. Solving a game takes this to the next level by replacing the heuristics with perfection.
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The game of chess as always been viewed as an iconic representation of intellectual prowess. Since the very beginning of computer science, the challenge of being able to program a computer capable of playing chess and beating humans has been alive and used both as a mark to measure hardware/software progresses and as an ongoing programming challenge leading to numerous discoveries. In the early days of computer science it was a topic for specialists. But as computers were democratized, and the strength of chess engines began to increase, chess players started to appropriate to themselves these new tools. We show how these interactions between the world of chess and information technologies have been herald of broader social impacts of information technologies. The game of chess, and more broadly the world of chess (chess players, literature, computer softwares and websites dedicated to chess, etc.), turns out to be a surprisingly and particularly sharp indicator of the changes induced in our everyday life by the information technologies. Moreover, in the same way that chess is a modelization of war that captures the raw features of strategic thinking, chess world can be seen as small society making the study of the information technologies impact easier to analyze and to grasp.
Lomonosov endgame tablebases
  • Convetka Ltd
Convetka Ltd. Lomonosov endgame tablebases. website : http://chessok.com/?page id=27966, 2013.