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Introduction to Connect-Four
We will introduce the game with two poorly-played games which demonstratesimple themes. As we will see, the strategy for the First_Player iscompletely different from that of the Second_Player. All of our games aretaken from a match between Miss Jacqueline Eques ('X') who always moves firstand Sir Hilary Knott ('O') who always moves second.Game 1: Mate on the Seventh Stone - combinatorial play We label the 42 cells on the board A1-G6 as shown. We have putthe Second Player's (Knott or O) moves in lower case.
Each player has made weak moves in this opening but in diagram 1-1,Knott's stones are scattered out-of-play while all four of Eques's stonesare about to participate in a quick forced win.
The game continues: Here are two more examples of combinatorial play you may wish to solvefor yourself. Eques can force four-in-a-row by her ninth stone in Diagram 1-3and by her eleventh stone in Diagram 1-4. Unlike the movesabove, these games were played in expert fashion, although there were twoerrors in the moves up to Diagram 1-3.Game 2: Second_Player gets Two Stones in the Second Row - positional play Unlike Game 1, where Eques's stones were poised for aquick 'combinational' attack based on immediate threats, in diagram 2-1the game will 'drag on' without incident into the ending. ButKnott is just as sure of victory as Eques was in diagram 1-1. All hehas to do is avoid playing in the first row. Later when X runs outof other moves, X will play in the first row, O will play above herand eventually complete a horizontal 4-in-a-row on the second row.Play the game out a few times until you're convinced that Knottvictory is inevitable.
In diagram 2-1, the unoccupied cells C2 and F2 (and B2 and G2as well) are called 'Threat cells' for Knott. We will call these'Minor Threats' since Knott doesn't yet even have 3-in-a-row and hewill need to complete two of the Threats to win. 'Major Threats' --whose occupation wins at once -- are of course better than Minor Threats.But from this diagram the Minor Threats are good enough to win.
'Positional play' is different from the 'Combinatorial'fireworks of Game 1. In Game 1 Eques fills in key threatening cellsand gets what she wants with a sequence (combination) of forcingplays. In Game 2, Knott's win will come in the ending and cannot behurried.
The 'Two Stones in the Second Row' are of value only tothe Second_Player. If Eques got this configuration it would becomparatively valueless. In the ending, as the columns are filled in, theFirst_Player will generally occupy the first, third and fifth rowson a column while the Second_Player gets the second, fourth andsixth rows. Eques therefore wants threats on an Odd Row.
Although in the opening Knott strives for two stones on the secondrow, it should be noted that the threats can actually be on any even rowif there is no odd-row Eques threat underneath them. Furthermore a singlestone may be enough if Knott also has a stone on the cell below: Knott will eventually win on the sixth row at a6-b6-c6-d6!
Eques has odd-row minor threats at C3 and E5, but you can verify thatEques cannot possibly obtain an odd-row threat on either the B- or G-column.The only possible odd-row threat in the A-column (A5) depends on both B4and C3 and there is no way Eques can ever develop this. With no tacticalchances and Odd Minor Threats in only two columns, Eques cannot counterattack.If one of Eques's threats were at C5 -- adjacent to Knott's threat -- a Drawmight be possible, but as it stands Knott will win readily with ordinarycaution.
The next example (or something very similar) arises constantly inactual play if Knott knows what he's doing and Eques doesn't. Here Eques hasno chance for any Odd threats at all and Knott can get seven-in-a-rowon the top row if he chooses. An inexperienced Eques player may be pleasedwith her Major Threats at B2 and F2, but these worthless threats willevaporate under Zugzwang in the ending.Let's look now in detail at an expert opening.
Game 3: The 3-4 Opening -- D4 is Poison [Joseki 13] Although the rightside is still empty of stones, Eques has greatinfluence there. She is threatening F1 which is quite strong and maybuild an attack along the C5-G1 diagonal as well. Let us consider thevarious possible Knott plays from diagram 3-1.
Knott plays A1 or B1 are too weak to consider.
If Knott defends with E1, Eques can play E2 with Sente, followed byE3 or B1 to win easily.
If Knott defends at F1, the variation might be: The Major Threat at E3 will win for Eques (even though Knotthas a Major Threat on the same cell). The plays at C5 and G1 couldhave been reversed. Knott F3 was intended to prevent third rowhorizontal threats; but D4 would have give Eques somewhat more trouble.
Knott can also put his fourth stone at C5: It is possible to analyze the endgame from diagram 3-3.Knott has the only Major Threat (E3) but this will be an irrelevant threat.Eques has odd-row threats on three columns: B5, E3, G3. (The minorthreat B5 is counted as the equivalent of a Major Threat because of theway threats E2 and E3 work together.) To be sure that Eques can win, wemust convince ourselves that the B5 threat will survive any Knottcounterplay on the leftside. The only way for Knott to cause troubleis if he gets to occupy both A4 and A5; then the Knott Threat at B4 willneutralize Eques's at B5. Obviously Eques can prevent Knott from gettingboth A4 and A5 so Eques's threats will survive. (Knott needs A4 sinceEques gets a winning Threat at B3 if Eques gets A4).
In other words, Eques's partial diagonal C4-D3 is 'undercut' byKnott's partial diagonal C3-D2 but that in turn is 'undercut' by theEques partial diagonal at C2-D1. Finding the bottom-most such diagonalthreat is an essential part of most endgame analyses.
In this variation both Eques and Knott played rather well, butthere were three errors. Eques should have played her fifth stone at E1.After Eques plays G1 Knott can force a draw with perfect play, and infact Knott can win after Eques's seventh stone at F3. (Eques should haveplayed D5 first.) Knott failed to win however; he should have played hiseighth stone at A1 instead of F5. Even as it was Eques had to play at A1and E1 precisely as she did to win.
Except for his eighth stone, Knott's defense was quite sound.(If F4 is omitted for example, Eques plays there and next at B1 or D5to get a positional win.)
If Knott plays D4 in diagram 3-1, Eques can play D5 and transpose into the '5-4 Opening'. But in that opening, Eques never gets an Even MajorThreat to go with her right-side odd threats and is forced to attack on bothsides of the board. In the 3-4 opening Eques can play F1, building a Threatat E2 immediately and winning with relative ease: In diagram 3-4, Eques will answer G2 with G3, or F4 with F5, andotherwise just take F4. Thus she gets Odd Minor Threats at E5 (andeither G5 or G3) with a win similar to the previous variation.Again, a possible Knott Threat at E3 is irrelevant.
Finally we let Knott play G1 as his fourth stone. This turns outto be his toughest defense. The plays at G2 and G3 correctly block enemy horizontalthreats. Eques E1 is a Sente (forcing) move since if Eques plays E2, Equeshas a fairly routine endgame win at her B5 Threat cell. Eques C5 was a Gotemove (it threatens nothing immediately) but takes the only importantcentral cell which can be taken 'for free'. (Taking D4 would givethe enemy D5 and E3 would give up E4.)
In the game through diagram 3-5, all of Eques's moves havebeen unconditionally forced (if she wants to guarantee victory) exceptfor her third and fourth stones at C2 and C4. Suppose she decides toplay C5 Gote before E1 Sente in the above line. Transposing the Gotestone C5 to be the sixth stone is losing play for Eques: In this variation, Eques E1 instead of D5 is ineffective,because Knott can answer E1 at F1 which is now strong enough for Knott.In diagram 3-6, Knott can now draw by playing at F1 or even win byplaying A1.
Returning to the main line (see diagram 3-5), what should beKnott's plan? He can't very well take a good cell like D4 or E3since Eques will get the even better cell above it (D5 or E4). Sincehe's forced to play on the periphery, he should make the most of it.If he can grab A1 and G4, followed by D4, he'll have both sides of theboard locked up with even threats. Of course Eques will not permitthis. In fact Eques must catch on to this plan at once (to avoid beingforced to take D4 later), and play A1 (or B1) in response to G4and specifically G4 in response to A1.
In the game, Knott took A1 though the ending might be similarafter G4 as well. Knott doesn't want to open up anywhere so he plays passivelyon the sixth row. Eques could of course have played E3 the turn beforewith equal effect.
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In the following Eques goes after Major Threats at F5, B3 and B5so most of Knott's moves are forced. We've interrupted the game briefly to indicate how hard itis for mere human Eques to always win. Wherever Knott now plays,Eques must answer in the same column! (Exceptions: F1 can also beanswered with D4. E6 must be answered at A2.)
The Eques threat configuration D4-D5-F5-F2 is a winner, with bothEques and Knott threats in the A column irrelevant. Eques has played passively at E6 (she could also have playedthere the turn before) to let Knott come to her. Eques has the winningtriple-odd threat configuration at B5-D5-F5. Knott's even threatsat B2-D4 would defeat Eques if viewed in isolation, but Eques also haseven threats at D4-F2 and Knott must now move (ZUGZWANG).
Knott has nothing to gain from letting Eques play F2 and then D5 sothe game continues: It is now fairly clear that Eques has a simple endgame win. Worthnoting is that with only four minor exceptions (shown on the Joseki chart),all of Eques's first 15 moves were forced in this game!
- E6 is as good as A5 on the passive thirteenth move.
- E3 could have been substituted for the passive ninth stone at G5.
- D4 gives an easier win than E4 as the fourth stone.
- The third stone can also be played at F1 or G1.
The moves through Diagram 4-2 constitute a Joseki: both sides haveplayed with excellent skill. Eques 2 A1 threatens to seize B2 quickly (eg,2 A1 d2 B1 c1 B2) and Knott c1 was the only way to prevent that. If Equesgot B2 it would establish the cells C3 and D4 as minor threats; since theseare central cells the configuration would be very strong for Eques. For onething Eques could claim D3 since Knott d3 conceding Eques D4 is out of thequestion.
Alternative Eques plays are 6 E5 and 2 B1 (no other deviations willguarantee victory.) Since Knott is destined to lose in any event againstexpert Eques play it isn't easy to make quantitative judgement about his play,but Knott 5 .. d2 and 2 .. e2 are alternatives which also require Eques toplay with care. Note that each of the plays e2-e4-d3-c5 were played whereEques was about to play to establish Odd Minor Threats.
Many of the Eques plays in this opening are easy to find. Eques 4 C4,5 E3 and 7 E5 seize valuable central cells for example. The play 8 A2 wouldbe easy to find; but Eques would have to foresee this to drop the stone 6 D2.
Eques's third stone should be played in the second row since allfirst row horizontal threats are resolved. 3 A2 seems 'out of the way',while 3 D2 d3 is unthinkable -- D3 is by far the best cell on the board.This leaves 3 C2 and 3 E2 as possibilities; by symmetry only the meaning ofEques's prior play at A1 is relevant.
Since Knott will need to answer an eventual Eques A3 with a4 in anyevent, the possibility of Eques A1-A2-A3 has no extra value. It appears thatthe only offensive significance of A1 is along the B2-C3-D4 diagonal.But Knott kills this diagonal by answering C2 at c3; hence 3 E2 may seemsuperior to 3 C2.
But as we have seen, Eques A1 has a hidden offensive meaning: Equesnow has the eventual option of dropping a stone at A2 which workstogether nicely with Eques stones C2-C4. True Knott may play a2 first,but this cell has no offensive value for Knott and would be a Gote play(lose a tempo) to boot.
The fact is that the 'Aji' of a possible future Eques A2 makes 3 C2the only winning play. 3 A2 and 3 G1 lose; other plays Draw. Here is aneasy notation to reflect this: With the alternative (Eques 3 E2) a position identical to Diag 4-1may be reached but with the stones at E2 and E3 having color reversed. In the short term, Diag 4-3 is tactically stronger than Diag 4-1:Knott must block the C4-F1 diagonal.
Except for the subtle error of Eques 3 E2, the moves throughDiagram 4-3 also constitute Joseki. To achieve the guaranteed draw, Equescan substitute only 5 D2, 5 E4 or 4 D2 and Knott can substitute only 4 ..d2. These moves are all logical and required. Knott 5 .. f1 failsbecause of the combination: Diagram 4-4 is very difficult to analyze, but it strongly favors Knott.Knott has the upper part of the board locked up with his Major Threats atb4 and f4, and either of these threats is enough for Knott victory if Equescannot counterattack. Conversely Knott can force a Draw with relative ease,for example: But 'the best defense is a good offense' and Eques's only chance --even for a draw -- is to go after her own winning threats. She has fivepotential odd-row threats (A3,B3,D5,F3,G3) and she will have to develop atleast three of these to have a chance. Because of the way Knott's diagonalsdominate the sides of the board, an Eques threat at A3 is almost worthlesswithout an Eques threat at B3, and G3 similarly depends on F3.
The cell D5 is absolutely essential for Eques (unless Knott carelesslyplays b3 or f3 before taking d4 D5). Eques must therefore defend now on theC5-F2-G1 diagonal or Knott will take f2 and either c5 or g1 and force Eques toplay D4 d5.
If Eques now takes F2, Knott's C5-G1 diagonal will disappear, while ifEques takes C5 or G1, Knott can take the other cell (Miai cell) and forceEques to expend another stone at F2. However Eques 8 F2 would be a weak play;this cell has no offensive significance for Eques and indeed undermines Eques'seventual threat at F3. Instead Eques should take G1 or C5 -- whichever isbetter. If Knott replies at the Miai cell and forces Eques to take F2, atleast Eques will have gotten the better of the two Miai cells.
But C5 and G1 also have minimal offensive value. C5 can contributeonly to the A3-B4-C5-D6 diagonal which is worthless unless Eques can undercutthe B-column at B3 (ie, occupy A2, A4 or E6). And G1 contributes to nopossible four-in-a-row except the unlikely-looking G1-G2-G3-G4.
The game may continue: After Knott makes the fine play of a2, Eques would need E6 to securethe B3 threat. Eques will thus need to build her attack on the right-handside.
But if she plays something like 10) C6 e5 B2 e6, Knott grabs theremaining E-column cells while Eques wastes a stone at B2. (In other words,Knott e5 was a Sente move threatening to consume the key cell D5.) Thus,Eques will probably start with E5: The exchange C6-b2 can be omitted from this game with a similar finalresult, but if Eques does play C6, Knott is almost in Zugzwang and the passiveplay at b2 is his only salvation. Eques needed three odd-threat columns towin the Zugzwang but she has only two: D5 and F3.
In Diag 4-6 Knott has only one possible play or Eques will win quickly. In Diag 4-7 Eques can play 15) D6 g3 A3 a4, but Eques Even Threat B4is worthless -- it is Knott who will inevitably play at b4 and win in theending. Eques's attack has simply fizzled out.
Referring way back to Diag 4-4 in which C5 and G1 appeared to beMiai cells, Eques might have played 8 G1 which Knott would answer at c5. Ifthe game then proceeded exactly as above (which it well might since the cellsC5 and G1 had minimal effect on the analysis) then a position ALMOST identicalto Diagram 4-7 is reached (Diag 4-8). But now Eques wins at once with adouble Atari, 15) G3 d6 G4. (After Eques 8 G1, Knott 9 ..a2 is no longerTesuji but is a subtle blunder; instead Knott must play to draw as mentionedabove.)
In the sequence 5) C4 d2 D3 f1 E4 b1 G1, all of the moves areuniquely forced if the players wish to guarantee their best result (Draw).Knott fails to win only because of Eques's Tesuji at 8 G1, which in turnderives its special value only from the tactical possibility of And this study arose from my failure to see the hidden Aji of Eques A1in Diagram 4-1. 3 E2 seemed to be a more logical move but I found Equesstruggling just to draw.
Who said Connect-Four is a trivial game?
If you decide to play 1 D1 e1 A1 as Eques, you will also want tomemorize Joseki 18. It is very pleasing: Knott's 2 .. e2 appears logical since it threatens to make the forcingmove e3 and also works towards getting Two Stones in the Second Row. Equescan prevail but she must play a precise sequence of forcing moves.
Usually Eques' goal is to get stones in the Third Row, but in thisunusual opening she plays in the second row with B2-C2-D2. Knott's plays atb3 and c3 are almost forced (Eques gets very strong threats if permitted tooccupy the Third Row) and there was of course no choice about a1 or d4.After this exchange, most of Eques' stones are out-of-play but the Equesconfiguration 'peeks' out from Knott's blanket at D3 and this is just enoughto win.
It turns out that Eques wins easily in this diagram: the threatformation on the rightside (D2-E3-F4-G5, D3-E3-F3-G3, along with the E4Major Threat) is victorious (it's similar to a 'J' but morecomplicated-looking); Knott's stones on the leftside may look impressivebut they can accomplish nothing. The simplest way for Eques to continue is Eventually Eques will get F4 and win the ending at G5. If this isn'tobvious, play the game out remembering to answer Knott g2 at G3.
Threat Analysis
When an ending is devoid of tactical complications a simpleanalysis will predict the outcome. This analysis is useful earlier inthe game to determine which potential threats are worth pursuing.Briefly:
Step 1) Analyze the Odd Threats of each player and classify the game asWon for Eques, Won for Knott or Drawn. Even Threats are relevant only whenin the same column as an Odd Threat.Step 2) Only if step 1 indicates a Draw, check for Knott Even Threats.Any such threat is then enough for Knott victory.
In more detail:
Assuming Knott does not get a counterthreat, an Eques Major Threat onan odd row (row 3 or 5) will normally be enough to win.Odd Minor Threats may be good enough if Eques has three of them inseparate columns. Just a pair of odd minor threats are enough to win if Eques hasan Even Threat in one of the same columns. The Even threat can be a Majorthreat or it can be half of a Mixed (Odd/Even) Minor threat pair. In the absence of Eques threats, Knott needs two Odd Major Threatsto win. He also wins with any Even Threat. As seen in Diagram 2-1, theEven Threat can be a Minor threat pair (but not a Mixed Threat), or evena nearly empty row (Diagrams 2-2, 2-3). But Knott's Even Threat is worthlesswhen Eques has winning Odd threats unless the Knott Threat undercuts (is onthe same column as and below) the Eques threats.
When both players have Odd Major Threat(s), the importantconsideration is the total number of columns with Odd Threats. With twosuch columns, either player needs threats in both of them to win. Eques hasthe advantage with one or three such columns; Knott has the advantage withfour. When both players have two Odd Major Threats Knott wins if the threatsoccupy two columns, Eques if three, the game is Drawn with the threats onfour separate columns.
In counting the odd-threat columns for the above rule, a Knott MinorThreat is counted only if it is a Mixed threat with the Even threat on thesame column as and below another odd threat (whether Eques's or Knott's).
When one column has two threats of the same player, they are ofspecial value only when adjacent (or a 2nd-row/5th-row pair.) If Knott hasMajor Threats on the 2nd and 5th rows of the same column, and Eques has anOdd Major Threat elsewhere the game is only Drawn; if a third column containsanother Odd Major Threat Knott wins if it is his, the game is still Drawn ifit is Eques's.
When one column has threats of each player, Eques dominates when boththreats are Odd (unless there are an even number of odd-threat columns),and Knott dominates when both are Even. If one threat is Odd and the otherEven, the lowermost threat dominates.
Here is a configuration which may arise from time to time. Knott will win at A2 eventually, because of his compound Threatsin the C-column. But he must be in no hurry to play c2. Instead he mustpreserve his c3 threat until Eques is in Zugzwang and forced to play F2. Eques has an Odd Major Threat at F3, but she would need a
second Odd Major Threat to prevail. Eques's adventures in the G-columnwere of course mistakes.
It is not so easy to analyze early positions but you will often beable to find the best moves if you understand the endgame objectives. Itis also easy to go wrong. For example: Both sides have played with skill so far. Eques could have grabbedOdd Minor Threats on the leftside with 8 D6, but victory would still requirecare and the play 8 G1 is just as good. Where should Eques play next?Eques may be tempted to play 9 G2 and establish F3 as an Odd MajorThreat. If Knott responds at b6 in an effort to Draw with the Counterthreatat c5, Eques can then take 10 F1 which kills ('undercuts') the Knott leftsidethreat.
But 9 G2 doesn't work. After 9 G2 f1! B6, Eques will discover thatthe Knott threat at c5 is still valid (refer to Endgame Principles II and III).Eques will eventually be in Zugzwang and forced to give up her Threat(18 C4 c5 F2 f3).
Instead Eques must play 9 F1 to destroy Knott's leftside, and whenKnott answers 9 .. g2, 10 C1 is the move to destroy Knott's even Threat atf4. Eques threatens C2 which establishes F5 as an Odd Major Threat at once.If Knott takes c2 first, Eques plays 11 E5 and F5 will still become a MajorThreat eventually, since Eques must get G5 or G6. But if Knott plays 10 .. f2in the diagram how should Eques respond?
If Eques responds at C2 or E5 to 10 .. f2, Knott will play 11 .. f3and undercut Eques' rightside threat with the Mixed Minor Threat f4-g5.The Eques partial diagonal D1-E2 is underneath Knott's partial diagonal d2-e3and this advantage must be preserved. In the diagram Eques must answerf2 at F3.
The following configuration (called the 'J') is seen very frequently.Usually it is good enough for Eques victory. When the G-column is eventually played, Eques will get G3 or G4. Ineither case F3 becomes an Odd Major Threat. Of course it is really thearrangement of the empty (Threat) cells -- shown here with '*'s -- thatprovides the victory.
Often Knott will prevent Eques from establishing this formation;when Eques plays E2, Knott takes e3. But sometimes Eques can build a 'J'by force. In that case Knott has only two hopes:
- to build counterthreat(s) on the opposite side of the board, or
- to 'cap' the 'J' with a Major Horizontal Threat:
There are other configurations which win in a fashion similar to the'J'. For example in Joseki 18, Eques wins when she gets this configuration: (Here there are four Minor Threat cells compared with three in the 'J'.But the effect is equivalent: since G3 and G5 are in the same column and bothon Odd Rows, they behave like a single Threat.)
The 'J' configuration is so common that you should memorize some ofthe variations that can arise. If Knott plays poorly the 'J' will win easily,but in these games Knott plays well.
Example 'J' Game 1)
Knott blocks the leftside 'J' by playing 5 .. c3, but he would losequickly if he played 6 .. e3 to block the 'J' on the right. He succeeds in'capping the J' with 7 .. e4, and the race is on as mentioned above. Asyou can see, Eques has won. Here Knott made two questionable moves. 3 .. d6is usually considered too passive. Having made that play, Knott shouldanswer 4 E1 at f1. Eques could still prevail, but she can no longer force aneasy win with 'J'.Example 'J' Game 2)
Knott could have played 5 .. e3 to stop the 'J', but Eques wouldanswer at E4 and, with Even Major Threat C2 already in the bag, would havea very strong attack. (If Knott responds to E4 on the leftside you canwin easily on the rightside: 7 G1 g2 G3 g4 G5 g6 E5; while if Knott playshis sixth stone at g1 the leftside provides victory: 7 B2 b3 B4 b5 D6.)Instead Knott plays 5 .. b2. This builds his own Odd Major Threatat c3 and he is threatening to get a Major Threat at c2 as well.Eques was not immediately concerned about this Double Threat, sinceafter 6 .. a2 E4 e5, the Knott c3 threat becomes irrelevant -- Knott wouldhave the leftside under control but no way to stop Eques from winning on therightside with her 'J'. The game continues When Knott takes e4, Eques must of course play A2 to prevent theDouble Threat. Similarly Knott must then play b3 or Eques will get anotherOdd Threat and Knott must respond B4 or Knott will get a Double Threat.Eques then goes about trying to build a second Threat at C5 -- Remember thatan Odd Threat cannot be 'undercut' by an enemy Odd Threat in the same column.
Eques has stones at B4 and D5 and when Knott takes b5, Eques takesD6 hoping that A3-C5 will be Odd Threats. This sequence arises frequently.Here it 'fails': Knott can block at a3 at once. But it doesn't really fail:Eques D6 is a much more powerful stone than Knott a3 as Eques demonstrateswhen she grabs E5. In the diagram, Eques has compound rightside threats(shown with the four '*'s) and will win. Eventually Knott will be forcedto play c2, f2 or g2. (Three columns have Odd Threats as required forEques victory according to the Endgame Principles.)
Example 'J' Game 3)
Here Knott tries 4 .. b2, blocks the leftside 'J' with 5 .. c3, butsuccumbs to a rightside 'J' due to the Eques Major Threat at E4. Eques cannow take her 'J' by playing E3 whenever she wants but it is often goodenough to grab it at once: Knott has 'capped' the 'J' and appears to be winning the race sincehe gets to drop a stone in the F-column before Eques can drop at G1. ButEques has a trick up her sleeve: Eques wins since she threatens Eques F5 with a winning Odd Threat atG5, and if Knott blocks at f5, Eques 13 G3 wins at once.I suggest you memorize these three games in which Eques has playedwell. Then it will be easy to win with 4 B1 as in Example Games 2 and 3 andyou have less need to master the complicated lines that arise after 4 E2.
In the next game Eques makes a mistake and loses despite her 'J'configuration.
Game 5: Counterthreat
Here is an endgame analysis based on the above rules. Thisformation should lead to a Knott win: Eventually the A column will be played and Eques will get A3 orA4. In either case, B3 becomes a Major Threat -- thus B3 can be treated asa Major Threat as it stands. Similarly G5 can eventually be promoted toa Major Threat for Knott. With two odd-threat columns and neither playerhaving threats in both columns, we have an Odd-Threat Draw. Knott will winthe game of course since he doesn't actually have to play up the F-column toenjoy the G5 threat: his Even Threat F4 will win the ending.Usually the Eques partial diagonal at D1-E2 will threaten F3-G4and will neutralize Knott threat G5. That does not apply here sinceEques can hardly play F3. The winning Knott threat configuration is a bitfragile, by the way, and Knott will get only a Draw (or even lose) unless heis careful.
Even without knowing the moves that led to diagram 5-1, a glancesuggests that Eques neglected to seize the key cell D5 when she had the chance. Eques has succeeded in building a 'J configuration' in the lower left,but she should have played 3 D5 instead. Knott is developing a huge pluralityin the upper center. Knott is even threatening to destroy Eques's 'J' byplaying up the B column (Eques cannot answer b2 with B3 since Knottwins at b4). To preserve the 'J' Eques must drive up the A columnimmediately: This is the threat configuration shown in Diagram 5-1. Where shouldKnott play next in diagram 5-3, by the way, or does it matter?
It appears that Knott can win quickly by playing at C6 and thenup the F column since F3 and F4 would both be winning cells. Butthis won't work: Knott gets his Double Threat at F3-F4 but Eques gets a DoubleThreat at B3-B4 and wins first. In diagram 5-3, Knott can win with anyplay except C6 or B2.
The tactical complications in diagram 5-3 may obfuscate the essentialstrategic considerations in diagram 5-2. Suppose that Eques reverses histenth and eleventh stones: Knott must now take care to grab the critical G-column cells beforeplaying even a single stone in the F-column: Eventually Knott will play f1 and there will be no stopping hisDouble Threat f4-f5. 13 g3 and 12 a5 also lead to Knott victory.
To avoid misleading the reader I should point out that Knott does haveother winning sequences besides those shown in diagrams 5-3 and 5-5. But thekey configuration shown in diagram 5-1 is ultimately Knott's target.
Problems
Solutions
Problem A
Eques must play G1 and go after the D4-G1 diagonal. If she triesto force the issue withshe loses to Knott's counterattack on the upper left side.When Eques does play 5 G1, a pretty continuation iswith Eques then able to force four-in-a-row by her tenth stone. Here Equesmust also play specifically 6 G2 and 7 E2 to win.
Problem B
Eques has a slight advantage on each side of the board.She must make her threats 'work together' to win. The Knott MajorThreat at F2 appears to neutralize the rightside, but Eques intends toplay C4 eventually and thereby kill the F2 threat. But first thingsfirst! Eques must start at B6 since if Knott plays there, Eques C4 isno longer playable. In some variations Eques will eventually win atB6-C6-D6-E6.Problem C
Eques can play up the A-column and then theC-column with a series of Sente moves, eventually winning on theC6-D5-E4-F3 diagonal. But Knott has a Major Threat at F4 and cantherefore foil Eques's plan by playing F1-F2-F3. It's a race andEques wins if she plays A1 immediately.Problem D
The third-row cells are generally the best cells on theboard. Good players avoid playing on the second row (especially at D2)if it gives their opponent the better cell above it. But fourth-rowand fifth-row cells are also valuable. Thus here the great value of E4cuts down on the value of E3. D4 is almost unplayable for eitherplayer since it gives up D5 (the best cell on the board after D3).Although it is 'Gote', C5 has good long-term strategic value and Equesshould play there. It is a 'free' play: C6 has little value.On the topic of 'avoiding the second row', consider the common openingin which Knott and Eques respectively did and didn't follow this advice.The strongest continuation is thenBoth Victor and I wasted many computer-hours on this position in afruitless search for Eques victory: the game is drawn.
Problem D2
After Eques C5 in Problem D, Knott takes A1. Knott hopesto get G4 followed by D4 which would 'lock up' both sides of the boardwith even-row threats. If Eques lets Knott get G4, Eques will have tothen take D4 whereupon Knott can get a draw by grabbing the key cell D5.Therefore Eques must answer A1 at G4.Problem E
Eques must play 8 G1! to draw against expert Knott play.Earlier Eques should have played 3 C2 to win. Knott will win easily withhis b4 and f4 threats unless Eques can build threats at B3 and/or F3. HenceEques wants to occupy cells like A2, A4, E6, etc but especially D5. But shecannot afford to play A2 now since then she loses the key cell D5:Thus Eques must play along the C5-G1 diagonal now to have any chance. Noneof these cells has much offensive value but, hard as it is to foresee, thepossibility of G1-G2-G3-G4 makes G1 the correct play and if Eques playsanywhere else Knott can win. This is explained more thoroughly in the text.Problem F
Eques must play A3 to Win. If Eques plays D2 eventually she willestablish Odd Minor Threats at A5 and C3 and the main virtue of A3 is toprotect this latent threat by preventing a Knott 'pushup' (eg, 5 B5 a3 A4 a5).A3 may seem out-of-the-way with the center up for grabs, but there areno prime cells available. As usual the preeminent value of D3 makes D2 a poorEques play despite the diagonal connection with B4 (5 D2 d3 A3 e2 is verystrong for Knott.) E2 is a good-looking play for Eques since it workstowards an Even Threat at C2 or C4. But against 5 E2 Knott will not waste atempo taking a5, as he would against 5 B5. Instead he will take e3 and (withb3 in the bag already) Eques would no longer have Third Row Horizontalpossibilities. (After E2 an expert continuation might bewith Eques finally blocking on the sixth row just to salvage a Draw.)After Eques makes the correct play of A3, one example continuation isFurther Notes on the Problems
Problem F
Note the bizarre leftside-heavy position after 5 B4 a3 A4 a5, eventhough the entire opening (except 5 B4) is expert. One good way to see the answer is to first realize that A3 and E2are the most valuable available cells, but that E2 'gives up' more -- E3is also very valuable, but A4 isn't. The opening position (empty board) is won for Eques, but 3 of the 4distinct 1-stone positions are won for Knott. Among 25 distinct 2-stonepositions, 14 are won by Eques, 5 by Knott and 6 are drawn.Among 121 distinct 3-stone positions, 18 are won by Eques, 74 arewon by Knott and 29 are Drawn. Details of this are shown next.
First Move Results
Results assume that all remaining moves are played as 'joseki' (best way forboth players). For example if Eques starts at D1 and Knott answers a1, Knottcan force victory if Eques plays B1, the game is drawn with best play if Equesplays A2, and Eques can win with any other play.There isn't room to show Knott's second stone in the lines where Eques hasblundered; the best play is often on the third row or in the D column.Of course there are many exceptions: In the variationKnott loses if he plays on the third row (f3) at either his 2nd or 3rd turn.
For common openings, continued best Eques play is shown on the next pageas 'Joseki.'
Opening Game Tree ('Joseki')
Another important Joseki begins with the movesThis opening is played so strongly by both players that we shall consider itin some detail. 1 .. e1 is probably Knott's best chance, since otherwiseEques can win by memorizing the key Joseki [7] - [12]. Similarly theplay 2 .. b2 is strong since it takes the powerful A1-E5 diagonal.Eventually Eques will take A1 or C1 and Knott will get the other point.Even though Knott owns the A1-E5 diagonal now, a diagonal pointing atthe key central cell D4 is worth far more than the diagonal pointingat B4. In fact Eques must seize A1 at once. For example in Joseki [23]:(Eques can substitute E2 as her 5th or 4th stone.) Since Knott must notlet Eques get E4, Eques effectively has E3 and E5 'in the bag' already.Although the rightside of the board is barren, Eques has establisheda winning configuration with the possibilities of C5-D5-E5-F5,D3-E3-F3-G3, C5-D4-E3-F2 and B1-C2-D3-E4. Eventually Knott will be inZugzwang and forced to give Eques D3 or F2 as well as E4 or E5.If Eques omits 3 A1, a possible continuation is 3 B3 d2 D3 a1with Knott winning.
Another variation is Joseki [25](Eques can substitute 10 A3, 8 G2, 6 C5, 5 C4 or 4 E2.) Since this issuch a stout defense by Knott, eleven stones are not enough for Eques toestablish an obvious triple threat, but her victory is inevitable, despiteKnott's counterthreat at d5, due to Zugzwang. Knott cannot try f1 and letEques get a quick win with the forcing moves D2-G3-D3-F3. EventuallyKnott will play at d2 for lack of anything better but his punishmentwon't stop: if he takes d4 he loses his counterthreat and temporizing atf1 fails because of Eques's tactical threat along the D4-G1 diagonal.
Eques can also win this opening by playing her fourth stone at E2instead of C2. This may be an interesting study. The Joseki chartabove gives some of Eques's forced continuations.
Comments on the openings
'Joseki' means 'expert opening' in Japanese. All the Eques(upper-case) moves win, and only these moves win.You may wonder how I chose which openings to give in the OpeningGame Tree. Strong Knott moves have been chosen, although space considerationspermit only a very small sample of the possibilities. But given the chosenKnott defense every possible winning Eques move is shown. This is ratherremarkable and means that Eques just barely has a won game.
Joseki [12] is the 'most forced' game in Connect-Four since Eques'sfirst thirteen plays are all uniquely forced. Even if Eques is content toDraw, the only permitted deviations are 13) anywhere; 10) F2; 8) E5; 5) B2;4) B1 (or F1); 1) C1.
While most of the Joseki were discovered by computer search,the moves in Joseki 1, 2 and 3 are a common human opening (the '5-4'Opening): if Knott allows Eques 8 B5, Eques has an easy win on the leftside.Similarly Knott has no chance without 5) ... b1, but Eques cannot play5) B1? e4 and give up the rightside. I spent several hours manuallyconfirming that Eques 9 B6 is required in the very frequently encounteredJoseki 2; this study prompted the design of the search software.
In Joseki 4 (another '5-4' Opening) Eques wins with a spectacularendgame race: she must play A1 before Knott plays f1. There are many otherKnott defenses possible in the '5-4' Opening. After practicing this openingto gain an appreciation of Eques's and Knott's objectives, you will do wellto play as in Joseki 7 or 8, where Eques wins with relative ease.
We cannot cover positional theory here. Briefly Eques needs Threat(s)on the 3rd or 5th rows. Without a Major Threat, she needs three threat cells;even-row threats are useful if there is an odd-row threat in the same column.
In Joseki 5, 7, 8 and 9 Eques wins with typical threat configurationsas shown in the diagrams. In Joseki 7, Knott takes Two Stones on the SecondRow, but Eques doesn't need the G-column so Knott threat F2 becomes worthless.In Joseki 8 Knott has Counterthreat C3 so Eques requires four threat cellsinstead of three. Eventually Knott will be in Zugzwang, forced to allow Equesto play F3 or G3 with a winning double-threat or C3 with an easy win. The cells B1 and F1 generally have more value than C1 or E1.(If Knott does not have D4, the cells A1 and G1 also have good value.)Hence Knott 3) ... b1 as in Joseki 9 and 10 may be a better try than 3) ... e1.But best for Knott is to try 3) ... d6! C1 b1! as in Joseki 12. This playmay appear unnatural since D6 is a remote cell of relatively small value,but here there is a mild Zugzwang operating in the first row.
Joseki 12 is played with utter precision and is a good study tosee the cells taken in the correct sequence. Eques never even gets a'winning endgame' -- Eques Threat F5 is undercut by the Knott d6-e5-f4-g3diagonal -- but she gets 'Super-sente' and in the end wins witha rare triple atari!
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Joseki 13 (a '3-4' Opening) is also played with utter precisionand is rather subtle. Neither player wants to take D4 and give up D5to the enemy. After 7) C5, Eques's stones dominate the board and with aneventual stone at E5, Eques gets a win similar to that shown in Joseki 6.
Joseki 15 is interesting because Eques never actually establishes awinning endgame; instead she uses 'Super-sente' to execute a rare combinationand in the diagram can play E3 and announce 'mate in two' with her twodouble-atari threats.
In Joseki 16 - 26, Knott disdains an early d2. In Joseki 18, Equescan hold off Knott's plurality in the upper left and will eventually win onthe barren rightside with Major Threat E4 and a plethora of minor threats.
Many other variations are possible after Knott 1 ... e1 or b1.Analysis is more difficult than after 1 ... d2 since the players jockey forposition in the corners before it is known who will get the commandingcentral cells D3, D4 and D5.
The games of a real match
Most of the example games were constructed by the author,but here are the six games of a real match between the author andJohn Tromp, another Connect-Four programmer. The author played firstin Games 1, 5 and 6. To add interest we agreed to start Games 3, 4,5 and 6 with the non-standard openings B1-b2-E1 or C1-b1-B2, openingswhich the computer search had revealed to result in Draws with optimalplay.The match was tied: 3 wins to John, 3 wins to the author andno Draws. To conform with our usual convention the First and Secondplayers will be called Eques and Knott.
Here are the moves of the games up to the resignations. Thesecond line shows what the result would have been with perfect playafter each move. The third line gives the optimal play(s) when anerror was made (including the opening 'errors' agreed to).
Comments on the Games.
In the opening diagonals which point towards central cellslike D4, D3 or even D2 are more valuable than the peripheral diagonalswhich D1 controls. Thus in
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An interesting position developed after 6 D4. Knott was afraidthat Eques would play A1 and establish an Odd Major Threat so 6 .. c2looked right -- it defended the leftside and established an EvenMajor Threat on the rightside. But, although Eques could no longerestablish an ordinary-looking 'triple threat', her huge plurality inthe upper leftside meant she could win easily as long she dropped a stoneat A2 or A3. (For example , 10 B6 a2 A3 g2 C4 a4 C5 a5 C6.)
Proper play for Knott would have been 6 D4 d5 A1 d6. NowKnott will presumably get e3 and e5 eventually and the equivalent ofa 'J configuration' on the rightside: he already has the Odd MajorThreat C3 and either f3 or f4 gives him a Major Threat at G3.
This was a game the author lost, but with hindsight the bestplays should be clear with a little study. C1 and F1 are worse thanuseless for either player. E6 is also almost worthless: it doesn'teven work towards a Knott Even Threat at C4 because of b3-B4. G1 isalso of minimal value. B3 and D3 are the 'prime real estate' availablebut they give up B4 and D4 which are even better. After the exchangeb3-B4, B3-b4, d3-D4 or D3-d4 there will be strong plays available, soeach player wants to time these plays for best effect.
If Knott answers 5 E5 with d3-D4-d5 it will be fairly clearto Eques to play to Draw. 5 .. b3 is also bad since B4 is such abetter cell. By process of elimination Knott should play a1 as he did;note that he's threatening to follow with b3 and establish a certainMajor Threat at C3. He's also threatening a2, after which eitherB3-a3-A4-b4 or A3-b3-B4-a4 give Knott a very comfortable left side.
By a similar process of elimination, 6 G1 is only the 'best'of six poor plays; Eques must answer a1 at B3, again with a drawishresult. Perhaps she was thinking to pick up G1-G2 with some rightsidestrength, but she should know from her own studies that she cannotsurvive such a dreadful 'double Gote' sequence.
Knott announced victory immediately when Eques blundered andmade the precise forcing moves d3-D4-b3-B4-g2. (Here only d5 for b3and a2 for d3 can be substituted.) Knott gets another Major Threat atF3, but doesn't really need it: the triple threat c2-c3-f5 might begood enough. Eques would have liked to have grabbed G2 for herself, butafter 8 G2 b4 B5 d5 C6 a2, Knott has a Double Major Threat.
Eques did make an error in the opening of Game 5 but it was not4 D2; it was 3 E2. A stone at D2 is preferred since it makes b3'off-limits' to Knott. Eques may have trouble winning if Knott grabsd3, but so will Knott and that is the important thing. Knott failed totake advantage: he should grab b3 in response to 3 E2, and if Eques thentakes B4, grab d2.
Consider the line Almost all of the results given have been established with exhaustivecomputer search. There are two exceptions, where a simple manual analysiswill suffice.
--- Knott wins after 1 A1 ------
Adopt the following strategy: Answer A1 at d1, A2 at a3, B1 or B3at b2 or b4, C1 at c2, E1 at e2, F1 or F3 at f2 or f4. Answer any othermove in the D-column. But if D6 is already occupied answer at F1 or F5 ifpossible, or in the B-column if c2 is occupied; F3 works if c2 is notoccupied.Following these rules you will eventually place a stone at an even-rowcell in the D-column or at an odd-row cell in the B- or F-column. At thatpoint you will have a relatively straightforward win and should 'switch offthe autopilot.'
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Another way to look at this strategy is that Eques will eventuallytake her choice of A4, B5, C3, D6, E3, F5 or G1 and then you strikeback. Of these D6 and F5 are Eques's most promising plays. If she takes F5and gives you d6, she will follow with F6 in an effort to establisha threat: Here Knott will eventually win at A6-B6-C6-D6. (Uncontested EvenThreats are so good for Knott that he prevails easily even though his threatsdo not fulfill the definition of even 'minor' threats.) The only possibleEques counterthreat is on the diagonal C3-D4-E5-F6. But if C1 c2 have alreadybeen played, Knott can play c3 and kill this diagonal at once, and if not,Knott simply avoids the plays c2 and e4, taking an odd-row cell in one ofthese columns instead.
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Eques may decide to take D6 and give you f3: Knott takes F3 and wins eventually on the D1-E2-F3-G4 diagonal.The only possible Eques counterthreat, the C1-D2-E3-F4 diagonal, is neutralizedsince Knott answers F4 at c1, or vice versa. If Eques had played C1 c2 beforeD6, Knott would answer in the B-column and not at f3.
--- Knott draws after 1 C1 ------
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Victor Allis established with a simple argument that Knott can guaranteeat least a Draw after 1. C1 d1. (He has a similar argument for 1. A1 d1and 1. B1 c1.) Knott simply plays above each Eques stone, playing in theC-column when Eques plays D6 or in the D-column when Eques plays C6. Sooneror later Eques must take D2-D4-D6 to block. The forced configuration is thuswhich is obviously a draw.To Draw when Eques opens at B1, answer B1 at c1, F1 at g1 or viceversa; play at any even-row cell when Eques plays in the sixth row andotherwise play above Eques's last play.
With either of these strategies do not be content with a Draw:look for an opportunity to grab a winning threat and 'switch off theautopilot.'
Glossary
- Aji
- a move which will eventually have Sente value. Consider the Eques stone at A1 in Joseki [17] which eventually allows A2.
- Atari
- a move which threatens to build Four-in-a-Row on the next move. Double Atari guarantees victory on the next move.
- Cell
- one of the 42 locations on the board where stones can eventually be placed.
- Counterthreat
- a threat which neutralizes an enemy threat which would otherwise win.
- Eques
- first player to move. Denoted by upper-case or 'X'.
- Error
- a move which is not Joseki. Denoted '?' in the game diagram.
- Even Threat
- a major or minor threat cell which is on the 2nd, 4th or 6th row.
- Four-in-a-Row
- four stones of the same color in a continuous line horizontally, vertically or diagonally. The object of the game.
- Gote
- a move which does not force the opponent's reply.
- J Configuration
- the common winning Eques configuration comprising D1-D3-E2-E3 or its mirror image.
- Joseki
- (proper play, an expert opening or sequence). Moves which are at least tied for best assuming both players continue to play optimally.
- Knott
- second player to move. Denoted by lower-case or 'O'.
- Major Threat
- the empty cell of an almost completed Four-in-a-Row. Unlike with Atari, the cell is not yet available for play. Vertical threats are not considered Major or Minor -- they can only become Atari.
- Miai
- two valuable cells one of which will be taken at once by Eques, the other by Knott or vice versa.
- Minor Threat
- one of two empty cells in a half-completed Four-in-a-Row.
- Mixed Threat
- one of a pair of minor threats of which one is Odd and one is Even. This can occur only along a diagonal.
- Odd Threat
- a major or minor threat cell which is on the 3rd or 5th row.
- Pushup Play
- a tactic where a player occupies cells like C1-C2-C3 to force his occupation of C5 (see Joseki [3]).
- Sente
- a move which forces the opponent's reply.
- Stone
- the colored marker which a player places at his turn.
- Tenuki
- a move which ignores the opponent's Gote move and plays in another quadrant. In Joseki [13] Eques 8 G4 appears to be Tenuki but is actually a necessary response as explained in the text.
- Tesuji
- a particularly fine move. Denoted '!' in the game diagram.
- Threat
- a cell a player hopes to occupy eventually as part of a Four-in-a-Row.
- Triple Odd Threat
- an eventual Eques victory provided by odd minor threats in three separate columns. This can arise in many ways. Odd threats in two columns are good enough if one is accompanied by a Mixed or Major Even Threat in the same column. (Exception: a threat pair using the 3rd and 6th rows has no value to Eques.)
- Undercut
- a threat which neutralizes an enemy threat above it.
- Zugzwang
- a move made when pass would be preferable. Most games are won in the ending after such a move by the loser. Zugzwang cannot occur in most games of the Tictactoe family, but the 'gravity rule' makes Connect-Four an exception.