Deep Blue versus Garry Kasparov

Deep Blue versus Garry Kasparov was a pair of six-game chess matches between world chess champion Garry Kasparov and an IBM supercomputer called Deep Blue. The first match was played in Philadelphia in 1996 and won by Kasparov. The second was played in New York City in 1997 and won by Deep Blue. The 1997 match was the first defeat of a reigning world chess champion to a computer under tournament conditions.

The 1997 match was the subject of a documentary film, The Man vs. The Machine.^[1]

Symbolic significance

Deep Blue's win was seen as very symbolically significant, a sign that artificial intelligence was catching up to human intelligence, and could defeat one of humanity's great intellectual champions.^[2] Later analysis tended to play down Kasparov's loss as a result of uncharacteristically bad play on Kasparov's part, and play down the intellectual value of chess as a game which can be defeated by brute force.^[3][4]

Nonetheless, a computer had successfully defeated a human champion, in a domain in which mastery was previously seen as a pinnacle of human intelligence. The dismissal of mastery of chess as a sign of true intelligence (whether fair or not) is an example of the so-called AI effect, the observation that the very fact that a computer can solve a problem causes the problem-solving involved to be classified as not true intelligence.^[5][6]

While there were eventually reasonable arguments that Kasparov could have won over the primitive brute force-based Deep Blue, if Kasparov had played up to his level, more sophisticated chess programs soon decisively outmatched humans, and the Deep Blue versus Garry Kasparov match is still cited as the symbolic turning point. This switched the canonical example of a game where humans outmatched machines to the ancient Chinese game of Go, a game of simple rules and far more possible moves than chess, which requires more intuition and is less susceptible to brute force.^[7] Go is widely played in especially China, South Korea, and Japan, and was considered one of the four arts of the Chinese scholar in antiquity. While Go programs were only able to defeat amateur players until 2015, in a 2016 match reminiscent of Deep Blue versus Garry Kasparov, Google DeepMind's AlphaGo program surprisingly defeated Lee Sedol in the match AlphaGo versus Lee Sedol.^[8] Where Deep Blue mainly relied on brute computational force to evaluate millions of positions, AlphaGo also relied on neural networks and reinforcement learning which more closely resemble human decision-making.

Summary

The 1996 match

Game #	White	Black	Result	Comment
1	Deep Blue	Kasparov	1–0
2	Kasparov	Deep Blue	1–0
3	Deep Blue	Kasparov	½–½	Draw by mutual agreement
4	Kasparov	Deep Blue	½–½	Draw by mutual agreement
5	Deep Blue	Kasparov	0–1	Kasparov offered a draw after the 23rd move.
6	Kasparov	Deep Blue	1–0
Result: Kasparov–Deep Blue: 4–2

The 1997 rematch

Game #	White	Black	Result	Comment
1	Kasparov	Deep Blue	1–0
2	Deep Blue	Kasparov	1–0
3	Kasparov	Deep Blue	½–½	Draw by mutual agreement
4	Deep Blue	Kasparov	½–½	Draw by mutual agreement
5	Kasparov	Deep Blue	½–½	Draw by mutual agreement
6	Deep Blue	Kasparov	1–0
Result: Deep Blue–Kasparov: 3½–2½

1996 match

Game 1

The first game of the 1996 match was the first game to be won by a chess-playing computer against a reigning world champion under normal chess tournament conditions, and in particular, classical time controls.

The game was played on February 10, 1996.

Deep Blue–Kasparov
1.e4 c5 2.c3 d5 3.exd5 Qxd5 4.d4 Nf6 5.Nf3 Bg4 6.Be2 e6 7.h3 Bh5 8.0-0 Nc6 9.Be3 cxd4 10.cxd4 Bb4 11.a3 Ba5 12.Nc3 Qd6 13.Nb5 Qe7 14.Ne5 Bxe2 15.Qxe2 0-0 16.Rac1 Rac8 17.Bg5 Bb6 18.Bxf6 gxf6 19.Nc4 Rfd8 20.Nxb6 axb6 21.Rfd1 f5 22.Qe3 Qf6 23.d5 Rxd5 24.Rxd5 exd5 25.b3 Kh8 26.Qxb6 Rg8 27.Qc5 d4 28.Nd6 f4 29.Nxb7 Ne5 30.Qd5 f3 31.g3 Nd3 32.Rc7 Re8 33.Nd6 Re1+ 34.Kh2 Nxf2 35.Nxf7+ Kg7 36.Ng5+ Kh6 37.Rxh7+ 1–0

Game 2

The second game began with the Open Catalan Opening. Kasparov played in what could be called a preemptive style blocking all Deep Blue's development attempts. The game lasted for 73 moves but eventually Deep Blue's operator had to resign the game for the computer in a position where both players had a bishop but Kasparov had three pawns to Deep Blue's one.

The game was played on February 11, 1996.

Kasparov–Deep Blue
1.Nf3 d5 2.d4 e6 3.g3 c5 4.Bg2 Nc6 5.0-0 Nf6 6.c4 dxc4 7.Ne5 Bd7 8.Na3 cxd4 9.Naxc4 Bc5 10.Qb3 0-0 11.Qxb7 Nxe5 12.Nxe5 Rb8 13.Qf3 Bd6 14.Nc6 Bxc6 15.Qxc6 e5 16.Rb1 Rb6 17.Qa4 Qb8 18.Bg5 Be7 19.b4 Bxb4 20.Bxf6 gxf6 21.Qd7 Qc8 22.Qxa7 Rb8 23.Qa4 Bc3 24.Rxb8 Qxb8 25.Be4 Qc7 26.Qa6 Kg7 27.Qd3 Rb8 28.Bxh7 Rb2 29.Be4 Rxa2 30.h4 Qc8 31.Qf3 Ra1 32.Rxa1 Bxa1 33.Qh5 Qh8 34.Qg4+ Kf8 35.Qc8+ Kg7 36.Qg4+ Kf8 37.Bd5 Ke7 38.Bc6 Kf8 39.Bd5 Ke7 40.Qf3 Bc3 41.Bc4 Qc8 42.Qd5 Qe6 43.Qb5 Qd7 44.Qc5+ Qd6 45.Qa7+ Qd7 46.Qa8 Qc7 47.Qa3+ Qd6 48.Qa2 f5 49.Bxf7 e4 50.Bh5 Qf6 51.Qa3+ Kd7 52.Qa7+ Kd8 53.Qb8+ Kd7 54.Be8+ Ke7 55.Bb5 Bd2 56.Qc7+ Kf8 57.Bc4 Bc3 58.Kg2 Be1 59.Kf1 Bc3 60.f4 exf3 61.exf3 Bd2 62.f4 Ke8 63.Qc8+ Ke7 64.Qc5+ Kd8 65.Bd3 Be3 66.Qxf5 Qc6 67.Qf8+ Kc7 68.Qe7+ Kc8 69.Bf5+ Kb8 70.Qd8+ Kb7 71.Qd7+ Qxd7 72.Bxd7 Kc7 73.Bb5 1–0

Game 3

In the third game Kasparov played the Sicilian Defence to which Deep Blue responded with the Alapin Variation. The game lasted for 39 moves and was drawn.

The game was played on February 13, 1996.

Deep Blue–Kasparov
1.e4 c5 2.c3 d5 3.exd5 Qxd5 4.d4 Nf6 5.Nf3 Bg4 6.Be2 e6 7.0-0 Nc6 8.Be3 cxd4 9.cxd4 Bb4 10.a3 Ba5 11.Nc3 Qd6 12.Ne5 Bxe2 13.Qxe2 Bxc3 14.bxc3 Nxe5 15.Bf4 Nf3+ 16.Qxf3 Qd5 17.Qd3 Rc8 18.Rfc1 Qc4 19.Qxc4 Rxc4 20.Rcb1 b6 21.Bb8 Ra4 22.Rb4 Ra5 23.Rc4 0-0 24.Bd6 Ra8 25.Rc6 b5 26.Kf1 Ra4 27.Rb1 a6 28.Ke2 h5 29.Kd3 Rd8 30.Be7 Rd7 31.Bxf6 gxf6 32.Rb3 Kg7 33.Ke3 e5 34.g3 exd4+ 35.cxd4 Re7+ 36.Kf3 Rd7 37.Rd3 Raxd4 38.Rxd4 Rxd4 39.Rxa6 b4 ½–½

Game 4

The fourth game was the second game to end in a draw, even though at one point Deep Blue's team refused Kasparov's draw offer. The opening played was the Semi-Slav Defense.

The game was played on February 14, 1996.

Kasparov–Deep Blue
1.Nf3 d5 2.d4 c6 3.c4 e6 4.Nbd2 Nf6 5.e3 Nbd7 6.Bd3 Bd6 7.e4 dxe4 8.Nxe4 Nxe4 9.Bxe4 0-0 10.0-0 h6 11.Bc2 e5 12.Re1 exd4 13.Qxd4 Bc5 14.Qc3 a5 15.a3 Nf6 16.Be3 Bxe3 17.Rxe3 Bg4 18.Ne5 Re8 19.Rae1 Be6 20.f4 Qc8 21.h3 b5 22.f5 Bxc4 23.Nxc4 bxc4 24.Rxe8+ Nxe8 25.Re4 Nf6 26.Rxc4 Nd5 27.Qe5 Qd7 28.Rg4 f6 29.Qd4 Kh7 30.Re4 Rd8 31.Kh1 Qc7 32.Qf2 Qb8 33.Ba4 c5 34.Bc6 c4 35.Rxc4 Nb4 36.Bf3 Nd3 37.Qh4 Qxb2 38.Qg3 Qxa3 39.Rc7 Qf8 40.Ra7 Ne5 41.Rxa5 Qf7 42.Rxe5 fxe5 43.Qxe5 Re8 44.Qf4 Qf6 45.Bh5 Rf8 46.Bg6+ Kh8 47.Qc7 Qd4 48.Kh2 Ra8 49.Bh5 Qf6 50.Bg6 Rg8 ½–½

Game 5

The fifth game was the turning point in the match. During the game, Kasparov, playing Black, chose a different opening, the Four Knights Game, from the Sicilian Defence he had played in games one and three and came out on top. This was particularly embarrassing for the Deep Blue team, because they had declined Kasparov's draw offer after the 23rd move. This was the only game in the match that Black won.

The game was played on February 16, 1996.

Deep Blue–Kasparov
1.e4 e5 2.Nf3 Nf6 3.Nc3 Nc6 4.d4 exd4 5.Nxd4 Bb4 6.Nxc6 bxc6 7.Bd3 d5 8.exd5 cxd5 9.0-0 0-0 10.Bg5 c6 11.Qf3 Be7 12.Rae1 Re8 13.Ne2 h6 14.Bf4 Bd6 15.Nd4 Bg4 16.Qg3 Bxf4 17.Qxf4 Qb6 18.c4 Bd7 19.cxd5 cxd5 20.Rxe8+ Rxe8 21.Qd2 Ne4 22.Bxe4 dxe4 23.b3 Rd8 (=) 24.Qc3 f5 25.Rd1 Be6 26.Qe3 Bf7 27.Qc3 f4 28.Rd2 Qf6 29.g3 Rd5 30.a3 Kh7 31.Kg2 Qe5 32.f3 e3 33.Rd3 e2 34.gxf4 e1=Q 35.fxe5 Qxc3 36.Rxc3 Rxd4 37.b4 Bc4 38.Kf2 g5 39.Re3 Be6 40.Rc3 Bc4 41.Re3 Rd2+ 42.Ke1 Rd3 43.Kf2 Kg6 44.Rxd3 Bxd3 45.Ke3 Bc2 46.Kd4 Kf5 47.Kd5 h5 0–1

Game 6

The sixth game was an illustration of just how badly computers can play in some positions. Employing anti-computer tactics and keeping the focus of the game on long-term planning, Kasparov slowly improved his position throughout the mid-game while Deep Blue wasted time doing nothing. By the end of the game, Deep Blue's pieces were crammed into its queenside corner, with no moves to make aside from shuffling its king. Kasparov had all the time in the world to finish the rout.^[9] Kasparov's next move would probably have been 44.Qe7 to exchange the queens. That would have allowed his pawn, which was about to promote, to advance.^[10]

The game was played on February 17, 1996.

Kasparov–Deep Blue
1.Nf3 d5 2.d4 c6 3.c4 e6 4.Nbd2 Nf6 5.e3 c5 6.b3 Nc6 7.Bb2 cxd4 8.exd4 Be7 9.Rc1 0-0 10.Bd3 Bd7 11.0-0 Nh5 12.Re1 Nf4 13.Bb1 Bd6 14.g3 Ng6 15.Ne5 Rc8 16.Nxd7 Qxd7 17.Nf3 Bb4 18.Re3 Rfd8 19.h4 Nge7 20.a3 Ba5 21.b4 Bc7 22.c5 Re8 23.Qd3 g6 24.Re2 Nf5 25.Bc3 h5 26.b5 Nce7 27.Bd2 Kg7 28.a4 Ra8 29.a5 a6 30.b6 Bb8 31.Bc2 Nc6 32.Ba4 Re7 33.Bc3 Ne5 34.dxe5 Qxa4 35.Nd4 Nxd4 36.Qxd4 Qd7 37.Bd2 Re8 38.Bg5 Rc8 39.Bf6+ Kh7 40.c6 bxc6 41.Qc5 Kh6 42.Rb2 Qb7 43.Rb4 1–0

1997 rematch

Game 1

The 1997 rematch began with the King's Indian Attack, which led Kasparov to victory in 45 moves.

This game was played on May 3, 1997.

Kasparov–Deep Blue
1.Nf3 d5 2.g3 Bg4 3.b3 Nd7 4.Bb2 e6 5.Bg2 Ngf6 6.0-0 c6 7.d3 Bd6 8.Nbd2 0-0 9.h3 Bh5 10.e3 h6 11.Qe1 Qa5 12.a3 Bc7 13.Nh4 g5 14.Nhf3 e5 15.e4 Rfe8 16.Nh2 Qb6 17.Qc1 a5 18.Re1 Bd6 19.Ndf1 dxe4 20.dxe4 Bc5 21.Ne3 Rad8 22.Nhf1 g4 23.hxg4 Nxg4 24.f3 Nxe3 25.Nxe3 Be7 26.Kh1 Bg5 27.Re2 a4 28.b4 f5 29.exf5 e4 30.f4 Bxe2 31.fxg5 Ne5 32.g6 Bf3 33.Bc3 Qb5 34.Qf1 Qxf1+ 35.Rxf1 h5 36.Kg1 Kf8 37.Bh3 b5 38.Kf2 Kg7 39.g4 Kh6 40.Rg1 hxg4 41.Bxg4 Bxg4 42.Nxg4+ Nxg4+ 43.Rxg4 Rd5 44.f6 Rd1 45.g7 1–0

Deep Blue's 44th move in this game puzzled Kasparov, and he attributed it to "superior intelligence". Allegedly, the move was a result of a bug in which Deep Blue, unable to determine a desirable move, resorted to a fail-safe. Nate Silver proposes that Kasparov "concluded that the counterintuitive play must be a sign of superior intelligence", leading him to lose the second game.^[11]

Game 2

In this game Kasparov accused IBM of cheating, a claim repeated in the documentary Game Over: Kasparov and the Machine.^[12][13] Kasparov eventually resigned, although post-game analysis indicates that the game could have been drawn. The game started with the Ruy Lopez opening, Smyslov Defence variation.

This game was played on May 4, 1997.

Deep Blue–Kasparov
1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.0-0 Be7 6.Re1 b5 7.Bb3 d6 8.c3 0-0 9.h3 h6 10.d4 Re8 11.Nbd2 Bf8 12.Nf1 Bd7 13.Ng3 Na5 14.Bc2 c5 15.b3 Nc6 16.d5 Ne7 17.Be3 Ng6 18.Qd2 Nh7 19.a4 Nh4 20.Nxh4 Qxh4 21.Qe2 Qd8 22.b4 Qc7 23.Rec1 c4 24.Ra3 Rec8 25.Rca1 Qd8 26.f4 Nf6 27.fxe5 dxe5 28.Qf1 Ne8 29.Qf2 Nd6 30.Bb6 Qe8 31.R3a2 Be7 32.Bc5 Bf8 33.Nf5 Bxf5 34.exf5 f6 35.Bxd6 Bxd6 36.axb5 axb5 37.Be4 Rxa2 38.Qxa2 Qd7 39.Qa7 Rc7 40.Qb6 Rb7 41.Ra8+ Kf7 42.Qa6 Qc7 43.Qc6 Qb6+ 44.Kf1 Rb8 45.Ra6 1–0

At the time it was reported that Kasparov missed the fact that after 45...Qe3 46.Qxd6 Re8, Black (Kasparov) can force a draw by perpetual check (threefold repetition). His friends told him so the next morning.^[14] They suggested 47.h4 h5!, a position after which the black queen can perpetually check White. This is possible as Deep Blue moved 44.Kf1 instead of an alternate move of its king. Regarding the end of game 2 and 44.Kf1 in particular, chess journalist Mig Greengard in the Game Over film states, "It turns out, that the position in, here at the end is actually a draw, and that, one of Deep Blue's final moves was a terrible error, because Deep Blue has two choices here. It can move its king here or move its king over here. It picked the wrong place to step." Another person in that film, four-time US champion Yasser Seirawan, then concludes that, "The computer had left its king a little un-defended. And Garry could have threatened a perpetual check, not a win but a perpetual check."

Modern chess engines consider the final position as better (but maybe not won) for white.^[15][16] The mistake was actually not 44.Kf1 but 45.Ra6?! when instead 45.Qd7+ was a clear win (White will exchange Queens, bring his King to e4 and his Bishop to g6 after which black sooner or later is in Zugzwang, and will be mated, or lose his b5 pawn, or the d5 pawn advances, opening roads for the white king).

Game 3

The third game was interesting because Kasparov chose to use an irregular opening, the Mieses Opening. He believed that by playing an esoteric opening, the computer would get out of its opening book and play the opening worse than it would have done using the book. Although this is nowadays a common tactic, it was a relatively new idea at the time.^[17] Despite this anti-computer tactic, the game was drawn.

This game was played on May 6, 1997.

Kasparov–Deep Blue
1.d3 e5 2.Nf3 Nc6 3.c4 Nf6 4.a3 d6 5.Nc3 Be7 6.g3 0-0 7.Bg2 Be6 8.0-0 Qd7 9.Ng5 Bf5 10.e4 Bg4 11.f3 Bh5 12.Nh3 Nd4 13.Nf2 h6 14.Be3 c5 15.b4 b6 16.Rb1 Kh8 17.Rb2 a6 18.bxc5 bxc5 19.Bh3 Qc7 20.Bg4 Bg6 21.f4 exf4 22.gxf4 Qa5 23.Bd2 Qxa3 24.Ra2 Qb3 25.f5 Qxd1 26.Bxd1 Bh7 27.Nh3 Rfb8 28.Nf4 Bd8 29.Nfd5 Nc6 30.Bf4 Ne5 31.Ba4 Nxd5 32.Nxd5 a5 33.Bb5 Ra7 34.Kg2 g5 35.Bxe5+ dxe5 36.f6 Bg6 37.h4 gxh4 38.Kh3 Kg8 39.Kxh4 Kh7 40.Kg4 Bc7 41.Nxc7 Rxc7 42.Rxa5 Rd8 43.Rf3 Kh8 44.Kh4 Kg8 45.Ra3 Kh8 46.Ra6 Kh7 47.Ra3 Kh8 48.Ra6 ½–½

Game 4

In this game Kasparov played the Caro-Kann Defence. Kasparov got into time trouble late in the game. The sub-optimal moves he played in a hurry may have cost him the victory.^[18]

This game was played on May 7, 1997.

Deep Blue–Kasparov
1.e4 c6 2.d4 d6 3.Nf3 Nf6 4.Nc3 Bg4 5.h3 Bh5 6.Bd3 e6 7.Qe2 d5 8.Bg5 Be7 9.e5 Nfd7 10.Bxe7 Qxe7 11.g4 Bg6 12.Bxg6 hxg6 13.h4 Na6 14.0-0-0 0-0-0 15.Rdg1 Nc7 16.Kb1 f6 17.exf6 Qxf6 18.Rg3 Rde8 19.Re1 Rhf8 20.Nd1 e5 21.dxe5 Qf4 22.a3 Ne6 23.Nc3 Ndc5 24.b4 Nd7 25.Qd3 Qf7 26.b5 Ndc5 27.Qe3 Qf4 28.bxc6 bxc6 29.Rd1 Kc7 30.Ka1 Qxe3 31.fxe3 Rf7 32.Rh3 Ref8 33.Nd4 Rf2 34.Rb1 Rg2 35.Nce2 Rxg4 36.Nxe6+ Nxe6 37.Nd4 Nxd4 38.exd4 Rxd4 39.Rg1 Rc4 40.Rxg6 Rxc2 41.Rxg7+ Kb6 42.Rb3+ Kc5 43.Rxa7 Rf1+ 44.Rb1 Rff2 45.Rb4 Rc1+ 46.Rb1 Rcc2 47.Rb4 Rc1+ 48.Rb1 Rxb1+ 49.Kxb1 Re2 50.Re7 Rh2 51.Rh7 Kc4 52.Rc7 c5 53.e6 Rxh4 54.e7 Re4 55.a4 Kb3 56.Kc1 ½–½

Game 5

In this game, the King's Indian Attack opening was played. As in the previous game, Deep Blue played a brilliant endgame that secured a draw, when it was looking as if Kasparov would win.

This game was played on May 10, 1997.

Kasparov–Deep Blue
1.Nf3 d5 2.g3 Bg4 3.Bg2 Nd7 4.h3 Bxf3 5.Bxf3 c6 6.d3 e6 7.e4 Ne5 8.Bg2 dxe4 9.Bxe4 Nf6 10.Bg2 Bb4+ 11.Nd2 h5 12.Qe2 Qc7 13.c3 Be7 14.d4 Ng6 15.h4 e5 16.Nf3 exd4 17.Nxd4 0-0-0 18.Bg5 Ng4 19.0-0-0 Rhe8 20.Qc2 Kb8 21.Kb1 Bxg5 22.hxg5 N6e5 23.Rhe1 c5 24.Nf3 Rxd1+ 25.Rxd1 Nc4 26.Qa4 Rd8 27.Re1 Nb6 28.Qc2 Qd6 29.c4 Qg6 30.Qxg6 fxg6 31.b3 Nxf2 32.Re6 Kc7 33.Rxg6 Rd7 34.Nh4 Nc8 35.Bd5 Nd6 36.Re6 Nb5 37.cxb5 Rxd5 38.Rg6 Rd7 39.Nf5 Ne4 40.Nxg7 Rd1+ 41.Kc2 Rd2+ 42.Kc1 Rxa2 43.Nxh5 Nd2 44.Nf4 Nxb3+ 45.Kb1 Rd2 46.Re6 c4 47.Re3 Kb6 48.g6 Kxb5 49.g7 Kb4 ½–½

If White plays 50.g8=Q then Black can force a draw by threefold repetition, starting with 50...Rd1+ and then 51...Rd2+.

Game 6

Before the sixth game, the overall score was even: 2½–2½. As in game 4, Kasparov played the Caro-Kann Defence. He then allowed Deep Blue to make a daring knight sacrifice which wrecked his defence and forced him to resign in fewer than twenty moves.

This game was played on May 11, 1997.

Deep Blue–Kasparov
1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nd7 5.Ng5 Ngf6 6.Bd3 e6 7.N1f3 h6 8.Nxe6 Qe7 9.0-0 fxe6 10.Bg6+ Kd8 11.Bf4 b5 12.a4 Bb7 13.Re1 Nd5 14.Bg3 Kc8 15.axb5 cxb5 16.Qd3 Bc6 17.Bf5 exf5 18.Rxe7 Bxe7 19.c4 1–0

See also

• AlphaGo versus Lee Sedol, similar but with Go
• List of chess games
• Game Over: Kasparov and the Machine
• Arimaa – Kasparov's loss to Deep Blue inspired the creation of a new game designed to be difficult for computers, yet playable with a chess set.

References

Bibliography

Further reading

• Hsu, Feng-hsiung (2002). "Behind Deep Blue: Building the Computer that Defeated the World Chess Champion". Princeton University Press. ISBN 0-691-09065-3. 
• Pandolfini, Bruce (1997). "Kasparov and Deep Blue: The Historic Chess Match Between Man and Machine". Fireside Chess Library. ISBN 9780684848525. 
• King, Daniel (1997). "Kasparov v. Deeper Blue: The Ultimate Man v. Machine Challenge". Batsford. ISBN 0-7134-8322-9. 
• Newborn, Monty (1996). "Kasparov versus Deep Blue: Computer Chess Comes of Age". Springer. ISBN 0-387-94820-1. 
• Gardner, Martin (2001). "Gardner's Workout, "Kasparov's Defeat by Deep Blue" (chapter 12)". A K Peters: 91–95. ISBN 978-1-56881-120-8.