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This article uses algebraic notation to describe chess moves.

In chess, a swindle is a ruse by which a player in a losing position tricks his opponent, and thereby achieves a win or draw instead of the expected loss. It may also refer more generally to achieving a win or draw from a clearly losing position. Although "swindling" in general usage is synonymous with cheating or fraud, in chess the term does not imply that the swindler has done anything unethical or unsportsmanlike. Indeed, the best swindles can be quite artistic, and some are very famous.

Contents 1 Classic Swindle Example 2 Stalemate 3 The weak back rank 4 Perpetual check 5 The surprise mating attack 6 Bishops of opposite colors 7 Material insufficiency 8 Building a fortress 9 Zugzwang 10 Multiple themes 11 See also 12 References 13 External links

The American grandmaster Frank Marshall, a gifted tactician, was notorious for his many "Marshall swindles."[1] Perhaps the most celebrated is Marshall-Marco, Monte Carlo 1904. In the diagram at left, White's position looks hopeless. Black's b-pawn is two moves away from queening, and White's awkwardly placed knight and rook seem to have no way to stop it. White could play 45.Rxc7+, but Black simply responds 45...Kb8. Many players would resign here, but Marshall saw an opportunity to try to save his game. He continued 45.c6! Now Black should play 45...bxc6!, but disdained it because White could then play 46.Rxc7+ Kb8 47.Rb7+! Kxb7 48.Nc5+, winning Black's rook and stopping Black's pawn from advancing. Black should have played this line, however, because he still wins after 48...Ka7 49.Nxa4 Bd4! (trapping the knight) 50.Kf3 Ka6 51.Ke4 Ka5 52.Kxd4 Kxa4 53.Kc3 Ka3 and Black's pawn queens after all. Instead, Marco played 45...Be5?, mistakenly thinking that this would put an end to Marshall's tricks. The game continued 46.cxb7+ Kb8 (46...Kxb7? 47.Nc5+ wins the rook) 47.Nc5! Ra2+ 48.Kh3 b2 49.Re7! Ka7 Not 49...b1(Q)?? 50.Re8+ Ka7 51.Ra8+ Kb6 52.b8(Q)+, winning Black's newly created queen. 50.Re8! c6! 51.Ra8+ Kb6 52.Rxa2! b1(Q) (see position at right) White's resources finally seem to be at an end, but now Marshall revealed his deeply hidden point: 53.b8(Q)+! Bxb8 54.Rb2+! Qxb2 55.Na4+ Kb5 56.Nxb2. Amazingly, Marshall has caught Black's pawn after all, and is now even a pawn up. He continued to play with great ingenuity, and even won the game.[2] (Analysis by Khariton.)[3]

Swindles can occur in myriad different ways, but as illustrated below certain themes are often seen.

One classic way of saving a draw in a losing position is by stalemate. In that article, see Gelfand-Kramnik,[4] the "Swindle of the Century" Evans-Reshevsky,[5] Pilnick-Reshevsky,[6] (scroll down to 12th comment, by "Resignation Trap," but note that Black's 26th move should be ...Rc7, not ...c7) and Reshevsky-Geller.[7]

Another famous Marshall swindle is Marshall-MacClure, New York 1923 (diagram at left).[8] (No. 15) Marshall, a rook down, played 1.Rh6! Rxh6 2.h8(Q)+! Rxh8 3.b5! Now Black is up two rooks, but the only way to avoid stalemate is 3...Rd7 4.cxd7 (threatening 5.d8(Q)+, forcing stalemate) c5?? 5.bxc6 Kb8 6.Kxb6, when White even wins. Decades later, someone pointed out an alternative draw with 1.Rg6! fxg6 2.h8(Q)+ Rxh8 3.b5 or 1...Re8 2.Rg8 Rb8 3.b5.[9] (No. 11)

In Tchigorin-Schlechter, Ostend 1905[10] (diagram at right), a game between two of the strongest players of the day, an unusual combination of stalemate and zugzwang enabled the great Schlechter to rescue a desperate position. Schlechter, in extreme time trouble, played 44...Qc7+! Tchigorin, thinking Schlechter had blundered, responded 45.Qb6+?, seemingly forcing the trade of queens. Schlechter's 45...Ka8!! forced an immediate draw: 46.Qxc7 is stalemate, and 46.Ka6 Qc8+! 47.Ka5 allows a draw with either 47...Qc7! (zugzwang), when White cannot make progress, or 47...Qc3+! 48.Ka6 Qc8+! with a perpetual check.

As the above game illustrates, being in time trouble is sometimes actually helpful to the prospective swindler, whose opponent may assume that a move is a time pressure-induced blunder rather than a trap. See also this account of a swindle by the late Czechoslovak-German GM Ludek Pachman, who deliberately wasted almost an hour on his clock in order to get into time trouble and thereby lull his opponent into being swindled.

In Kasparov-McDonald, simultaneous exhibition, Great Britain 1986,[11] the world champion had a winning advantage, which he could have converted after (for example) 54.Qd6+ Kg7 55.c6. Instead, he played 54.Bxe4??, allowing 54...Rxg3+! 55.Kxg3 Qe5+! 1/2-1/2, since the forced 56.Qxe5 gives stalemate (diagram at right). Note that 55.Kh4!? (instead of 55.Kxg3), with the strong threat of 56.Qh7#, would have been met by 55...Rg4+! 56.Kxg4 (forced) Qd7+! 57.Qxd7 with a different stalemate.

Other swindles based on stalemate include Congdon-Delmar, New York 1880[12]; Post-Nimzowitsch, Barmen Masters 1905[13]; Schlechter-Wolf, Nuremberg 1906[14]; Janowski-Grünfeld, Marienbad 1925[15]; Bernstein-Smyslov, Groningen 1946[16]; Horowitz-Pavey, U.S. Championship 1951[17]; Fichtl-F. Blatny, Czechoslovakia 1956[18]; Portisch-Lengyel, Málaga 1964[19]; Matulović-Suttles, Palma de Mallorca Interzonal 1970 [20]; Fuller-Basin, Michigan Open 1992[21]; Boyd v. Glimbrant, Alicante 1992; [22]; and Pein-deFirmian, 1995 [23].

Mating threats along the opponent's back rank often enable one to win or draw from a lost position. An example is seen in Rhine-Nagle, U.S. Masters 1997 (position at above left). Black, a pawn up with White's king in danger, decides that it's time to finish White off, forgetting that even a losing position may have defensive resources. Black continued 26...Rg5 27.Rhg1 Ra2?? A powerful-looking move, threatening mate in two, but it actually loses by force. Black could have kept a winning position with (for example) 27...Qf4+ 28.g3 Qf2+ 29.Rg2 Qf7, or 28.Kh1? Rg3 29.Qd1 Raa3 (threatening Rxh3+) 30.Qf1 Qh4 (renewing the threat) 31.Kh2 g6! (31...Ra2 32.Qf5! is weaker) and now Black is winning after 32.Ra1 Qxb4; 32.Rb2 Rgd3; or 32.Rc1? Ra2! (threatening mate on h3) 33.Kh1 Raxg2! 28.d6+ Kh8 The only reasonable move. White wins after 28...Qf7 29.dxc7! Ra8 30.Rgd1! Qxb3 31.Rd8+ Kf7 32.Rxb3 or 28...Kf8 29.Rbf1! Rxg2+ 30.Rxg2 Qxf1 31.Qxa2. 29.Qxa2!! Qxa2 30.dxc7! (diagram at above right) Turning the tables: Black, although ahead a queen for a rook, is suddenly helpless against White's passed pawn on the seventh rank. Qc2 29...Qa8 30.Rbd1 Rf5 31.Rd8+ Rf8 32.Rgd1, and 29...Qg8 30.Rgd1 Rf5 31.Rd8 Rf8 32.Rbd1, also win for White. 30.Ra1! The threat of a back-rank mate decides the game. 30.Rbc1? Rxg2+! 31.Rxg2 Qxc1 32.Ra2! Qf4+ 33.Kg2 Qg5+ would allow Black to draw by perpetual check. h6 If 30...Qxc7, 31.Ra8+ and mate next move. The game concluded: 31.Ra8+ Kh7 32.c8(Q) Qe4 33.Qg8+ Kg6 34.Rf8 1-0 (Analysis by Fritz 8.)

In de Firmian-Shirazi, U.S. Championship 1986[24] (diagram at above left), GM de Firmian is ahead three pawns, normally an easily winning material advantage at this level. IM Shirazi played 27...Qg6! attacking White's rook and inviting White to take another pawn. GM Robert Byrne, annotating the game in the New York Times, noted that de Firmian could have consolidated his great material advantage with 28.Rb2! Re8 29.Bd2! (29.Be3? Rxe3! 30.fxe3 Qg3! forces White to take perpetual check with 31.Qe8+ Kh7 32.Qh5+ Kg8 33.Qe8+.).[25] Instead, he took the bait with 28.Rxb5?? Rxf2 Now Black threatens 29...Rf1#, and White loses his queen after either 29.Kg1 Bh2+! or 29.Be3 Rf1+ 30.Bg1 Rxg1+! 31.Kxg1 Bh2+. White tried 29.Qa8+ Rf8 30.Rg5 (diagram at above right), but now 30...Qe4!, the "marvelous Marshall masher" (Byrne), ended the game, since 31.Qxe4 allows 31...Rfl#.

In Zukertort-Steinitz, London 1883[26], the world champion, playing Black, had a difficult, possibly lost, game an exchange down, with his centralized king subject to attack by Black's queen and two rooks. He played 27...Qd4+! (driving White's king to the corner) 28.Kh1 Rxf4! White cannot capture the rook because of the back-rank mate with Qd1. He could retain his large advantage (according to Fritz 8) with a rook move along the first rank, such as 29.Rb1, when 29...Qxb2!? could be met by 30.Re3+. Instead, White played 29.Re1+ Re4 30.Rxe4+?? Qxe4 31.Qxa7 (diagram at right), intending 31...Qe1+ 32.Qg1. However, Steinitz slammed the door shut with 31...b6! when the only way White could have stopped the back-rank mate was by giving up his rook (e.g. 32.Re3 Qxe3 33.h3), leaving him a knight down. Zukertort resigned.

Draw by perpetual check is another oft-seen way of swindling a draw from a lost position. The position at left is from Ivanchuk-Moiseenko, Russian Team Championship, Sochi 2005.[27] Black is down two pawns against the world's sixth highest-rated player. Worse, Ivanchuk's pieces dominate the board. IM Malcolm Pein notes that after almost any sensible move, for example 30.Qc2, Black would be completely lost.[28] White would then threaten 31.Rd6 pinning the knight to the queen, and neither 30...Nf6 31.Bxf6 gxf6 32.Qxh7# nor 30...Nc5 31.Ree7 is an adequate response. 30.Qc2 would also guard against a possible ...Qd1+, the significance of which becomes apparent after seeing the game continuation.

Moiseenko met Ivanchuk's 30.Rb7?? with 30...Nf8!! This not only threatens 31...Nxe6, but also enables Black to meet 31.Rxb8 with 31...Qd1+ 32.Kh2 Qh5+ 33.Kg1 Qd1+, drawing by perpetual check. The perpetual check is based on White's weak back rank combined with his slightly compromised king position (no h-pawn). Note how pieces that are well-placed for attacking purposes may be misplaced for defensive purposes. White's rook on e6 was well placed when White had the initiative, but is of no use in stopping the threatened perpetual check. (Similarly, in Rhine-Nagle, Black's rook on g5 was an excellent attacking piece, but was poorly placed to defend Black's black rank or stop White's passed c-pawn.)

White tried 31.Rh6, but could not avoid the perpetual: 31...Rxb7 32.Qxb7 Qd1+ 33.Kh2 Rh5+ 34.Rxh5 34.Kg3!? (hoping for 34...Rxh6?? 35.Qxg7#) is met by 34...Rg5+! and White must repeat moves with 35.Kh2! Rh5+, since 35.Kh3?? Qh1#; 35.Kh4?? Qg4#; and 35.Kf4 Qg4# all get mated. 34...Qxh5+ 35.Kg3 Qg5+ 36.Kf3 Qf5+ 1/2-1/2 since White cannot escape the perpetual check.

Sometimes perpetual check can even save the draw in a very simplified ending. In Keres-Eliskases, Noordwijk 1938 [29] (diagram at left), Black seems to be in desperate straits: he can win either of White's pawns, but then the other will queen, leaving White with a theoretical win in the queen versus rook ending. However, the players agreed to a draw after 56...Rb6+! 57. Kc1 Rh6! Because of continuous checks and mate threats from Black's rook, White will never have time to queen either pawn. For instance, 58.Kd1 Kd3 59.Ke1 Ke3 60.Kf1 Kf3 61.Kg1 Rg6+! 62.Kh2 Rh6+! 63.Kg1 Rg6+ 64.Kf1 Rh6! 65.Ke1 Ke3 66.Kd1 Kd3 67.Kc1 Kc3 68.Kb1 and now Black even has a choice of draws: (a) 68...Rb6+ 69.Ka2 Ra6+! or (b) 68...Rh1+ 69.Ka2 Rh2+ 70.Ka3 Rh1! 71.Ka4 Kc4 72.Ka5 Kc5 73.Ka4 (forced) Kc4 etc.

A surprise mating attack is another way to swindle a win or draw from a lost position. In Karpov-Csom, Bad Lauterberg 1977[30] (diagram at left), GM Csom has thoroughly outplayed the reigning world champion, and is a knight and pawn ahead with a completely winning position. Karpov's last move, 49.Rd7 (from d1), looks to be a last gasp before resigning. Csom played the solid-looking 49...Nf8??, saving the attacked knight and attacking White's rook. But after Karpov's 50.Nf5!! (diagram at right), Csom resigned. White threatens both 51.Rh7+! Nxh7 52.Qg7# and 51.Qh2+! Kg8 (51...Nh4 52.Qxh4+ does not help) 52.Qg3+ followed by Qg7#, and there is no way to stop both threats. 50...Nxd7 and 50...exf5 stop the former threat, but not the latter. Tim Krabbé considers 50.Nf5!! one of the most fantastic moves ever played.[31]

Instead, Csom could have won with 49...Ng5! Now 50.Nf5!? would be met by 50...exf5 51.Qh2+ Kg8 52.Qh6 Re1+ 53.Kh2 (53.Kf2 Qf3#) 53...Rh1+! 54.Kxh1 Nf4+ 55.Rd5 Nxd5 and wins. If instead 50.Nh5!? Rg8 51.Nxf6 (or 51.Rg7 Nh4) Nh4! threatening 52...Qg2# (note that White cannot force mate with 52.Rh7+, since 52...Nxh7 gives discovered check by the rook).

Karpov perpetrated another such swindle in the 17th game of his 1978 world championship match against Viktor Korchnoi.[32] Bent Larsen wrote in his book on the match that Korchnoi "lost a position it seemed impossible to lose." Korchnoi has had a large advantage for most of the game, which Karpov has been desperately trying to draw. From the position at left, Yasser Seirawan in his book Winning Chess Brilliancies (p. 90), recommends 34.Re7 followed by Rd1+, when Korchnoi "still could have hoped to keep Black's forces from coordinating." Instead, Korchnoi played 34.Rf4+?!, which Karpov met with the surprising 34...Ne4!, giving up his last pawn. Now Seirawn notes that after 35.Rxh7 Nd2!, Black would intend ...Rxa4 and ...Ke3 to harass White's king, and White's rook on f4 would be awkwardly placed for defense. Instead, Korchnoi played 35.Rd7+ Ke3 36.Rf3+ Ke2 37.Rxh7 Ncd2! A diabolical move, actually encouraging Korchnoi to hold onto his a-pawn. 38.Ra3?! Seirawan suggests that Korchnoi could have safely forced a drawn position with 38.Rhf7 Rxa4 39.h3 Nxf3+ 40.Rxf3. 38...Rc6! Now Seirawan recommends 39.g3! Nf3+ 40.Rxf3 (rather than 40.Kg2 Ne1+ 41.Kh1 Rb1) 40...Kxf3 41.Rf7+, again with a drawn position. Instead Korchnoi, in time trouble, played the natural 39.Ra1?? Nf3+!! 0-1 (diagram at right) A horrific end: Black mates after 40.Kh1 Nf2 or 40.gxf3 Rg6+ 41.Kh1 Nf2. This game was critical to the outcome of the match, since Karpov won by the narrowest possible margin: 6 wins to 5, with 21 draws.[33]

The late British grandmaster Tony Miles was an accomplished swindler. His game against the Tunisian IM Slim Bouaziz from the 1979 Riga Interzonal[34] (see diagram at above left), is a fine example of using a surprise mating attack to swindle a win from a lost position. The game shows a subtle psychological build-up to a swindle by the swindler, and deadly overconfidence (or perhaps zeitnot) by the "swindlee." Bouaziz has completely outplayed Miles, and is on the verge of a major upset. Bouaziz is up a rook for a bishop and has a simple plan: queening his c-pawn. White's king is a little drafty, but seems to be well-defended by White's queen and pawns clustered around it. The game continued 40...Rh1 Shuffling about aimlessly with his rook, or so it seems. Now 41.Rxh5! really would have left Black with a hopeless position, but White didn't see the need. 41.c6 h4! Of course, the pawn is immune (42.Qxh4?? Qg1#). White sees that on 42.c7, he has to worry about 42...Rxh3!? 43.Kxh3 Qh1+ 44.Qh2 Qxf3+. He could still win with 45.Kxh4 Be7+ 46.g5 Qe4+ 47.Kg3! Qe3+ 48.Kg2! Qe4+ 49.Kf1! Qf3+ 50.Rf2!, but understandably prefers to avoid such complications. 42.Rcd2! Now White can meet 42...Rxh3? with 43.Rd1! Rg3+ 44.Qxg3 and wins. 42...Rc1 43.Rc2 Qb1! Now 44.Rxc1 Qxc1 would leave White hard-pressed to both save his c-pawn and protect his king against a possible perpetual check. 44.Rdd2! Rh1! (see diagram at above right) The rook returns to h1; White, his pawn on the verge of promotion, sees nothing to fear. 45.c7?? 45.g5!, giving White's king a flight square, would still have won easily. 45...Rxh3!! White suddenly is in deep trouble, with Black threatening 46...Qh1#. Had White appreciated the danger, he could still have drawn with 46.Qf1! Rg3+ 47.Kf2 Rxf3+ 48.Kxf3 Qxf1+ 49.Ke4 Qe1+ and with White's pawn so far advanced, Black must be content with a draw by perpetual check (analysis by Crafty). Not realizing the seriousness of his predicament, White played 46.Kxh3?? Qh1+ 47.Qh2 Qxf3+ 48.Kxh4 Be7+ 49.g5 49.Kh5 g6+ 50.Kh6 Qe3+ forces mate. 49...Bxg5+! 0-1 Too late, Bouaziz sees 50.Kxg5 f6+ 51.Kh4 g5#! (or 51.Kg6 Qg4#!).

David Bronstein, in his immortal losing game, valiantly but unsuccessfully tried to swindle Bogdan Śliwa with a surprise mating attack.

Sometimes a player who is behind in material can escape into an endgame with bishops of opposite colors, i.e. where one player has a bishop that moves on white squares and the other player a bishop that moves on black squares. In such endings, the superior side is often unable to win with two or even three extra pawns. An example of a swindle based on securing bishops of opposite colors is seen in the diagram at above left. Grandmaster Mark Taimanov, playing White, has a winning position because his bishop pair is very strong, his king is more active, and Black's pawns are weak. White could win with 1.Bc2! Be8 2.Bxb8 Kxb8 3.Ke5 followed by Kf6, winning Black's g-pawn and the game, or 1...Na6 2.Bd6. (Shamkovich and Schiller, Saving Lost Positions, p. 86) Taimanov saw this line, but thought that the order in which he played Bc2 and Bxb8 didn't matter. He played 1.Bxb8?, expecting to transpose into the above line after 1...Kxb8 2.Bc2 Be8 3.Ke5. Bronstein surprised him with 1...c5+!! 2.Kxc5 Bxa4, resulting in a dead-drawn bishops of opposite colors ending. After White moves his en prise bishop, Black can play 3...Kb7 followed by 4...Bc2; then Black can keep his king on b7 forever, blocking White's pawn, and shift his bishop along the b1-f5 diagonal to defend his own pawn.

The Dutch grandmaster Jan Hein Donner swindled future World Champion Bobby Fischer in similar fashion. In the position at above right, Fischer is a pawn ahead and could increase his advantage with 30.Qb1! threatening a decisive gain of material with 31.Bxf7+! Kxf7 (or 31...Qxf7 32.Rxc8+) 32.Qb7+ (or even 32.Rc7+ Rxc7 33.Qxf5). 30...Rxc4? would lose material to 31.Qb8+. (Mortazavi, The Fine Art of Swindling, p. 49) Instead, Fischer played 30.Bd3? expecting to gain a second pawn after 30...Qd7 31.Rxc8+ Qxc8 32.Bxa6. However, Donner crossed his plans with 30...Rxc2! 31.Bxf5 Rc1, regaining the queen and forcing a drawn pawn-down bishops of opposite colors ending. Fischer agreed to a draw after 32.Qxc1 Bxc1 33.Kf1 Kf8 34.Ke2 h6.[35]

Sometimes a player who is behind in material may achieve a draw by exchanging off, or sacrificing for, all of the opponent's pawns, leaving a position (for example, two knights versus lone king) where the superior side still has a material advantage but cannot force checkmate. (Properly speaking, this may or may not be a "swindle," depending on whether the superior side missed a clear win earlier.) The inferior side is also sometimes able to achieve an ending that is theoretically still lost, but where the win is difficult and may be beyond the opponent's abilities (for example, bishop and knight versus lone king;[36] queen versus rook;[37] two knights versus pawn, which is sometimes a win for the knights;[38] or two bishops versus knight[39][40]).

The diagram at left shows a simple example of forcing a draw by material insufficiency. Black, although two pawns down, draws easily with 1...Bxa3![41] Then 2.bxa3 is a standard book draw, since White's bishop is of the "wrong color" from the rook pawn (i.e., it moves on the squares opposite in color to that of the pawn's queening square) and thus can never drive Black's king from the a8 corner. 2.b3 (stopping 2...Bxb2!) avoids that scenario, but Black still draws because White has no pieces that can drive Black's king from the light square b7, blocking the pawn. Black can mark time with his bishop indefinitely, so his king cannot be driven off by zugzwang.

White drew in similar fashion in Parr-Farrand, England 1971. From the diagram at right, play continued 1.Rd5 Bf6 2.Rxf5! On 2...gxf5 3.Kf4, White's king will capture Black's f-pawn, then run back to h1, reaching a bishop and opposite-colored rook pawn draw. Instead, Black tried 2...Ke7 3.Rb5 Ke6, "but he soon had to admit that the draw was inevitable." (Van Perlo's Endgame Tactics, Second Edition, p. 270, No. 635)

Schmidt-Schaefer, Rheinhessen 1997 (diagram at left), is another straightforward example. Black has connected passed pawns, but if White can sacrifice his knights for them he can reach the drawn two knights versus lone king ending. Thus, 50.Nfe4! threatened to capture both pawns with the knights. 50...dxe4 51.Nxe4 Kd5 52.Nxc5! would also achieve that goal. Black tried 50...d4, but agreed to a draw after 51.Nxc5+ Kd6 52.Nb5+! Kxc5 53.Nxd4! (Analysis from ChessBase 9, Mega 2004 database.)

The three examples above arguably are not true swindles, but rather the inferior side's exploitation of a defensive resource available in the position. However, Chandler-Susan Polgar, Biel 1987[42] (diagram at right) is a bona fide swindle. Polgar has just played 53...Nh6!? (from g8), transparently playing for a rook pawn and wrong-colored bishop draw. GM Chandler obligingly played 54.gxh6+??, expecting 54...Kxh6 55.Kf6! when he will win because Black cannot get her king to h8. Polgar, however, responded 54...Kh8! with the standard draw. White's possession of a second h-pawn is immaterial, and the game concluded 55.Bd5 Kh7 56.Kf7 Kh8! 1/2-1/2

The position at left, the conclusion of a chess problem by the American master Frederick Rhine, provides a more complicated example of forcing a draw by material insufficiency. White draws with 5.Nxc4+! Nxc4 If 5...Kc6 6.Nxb6 Kxb6 7.Rxb2+, White's rook draws easily against Black's knight and bishop. 6.Rxb6+ Now Black's best try is 6...Kd5! or 6...Ke7!, when the endgame of rook against two knights and a bishop is a theoretical draw, as in Karpov-Kasparov, Tilburg 1991[43] and confirmed by the Shredder six-piece database.[44] The more natural 6...Nxb6+ leads to a surprising draw after 7.Kd8! (diagram at right), when any bishop move stalemates White, and any other move allows 8.Kxe8, when the two knights cannot force checkmate.

Building a fortress is another method of saving an otherwise lost position. It is often seen in the endgame, for example in endings with bishops of opposite colors, as discussed above.