Expert Play in Connect-Four

James D. Allen
19785 Stanton Ave
Castro Valley CA 94546
(c) Copyright 1990
All Rights Reserved.

Introduction to Connect-Four

We will introduce the game with two poorly-played games which demonstrate simple themes. As we will see, the strategy for the First_Player is completely different from that of the Second_Player. All of our games are taken from a match between Miss Jacqueline Eques ("X") who always moves first and Sir Hilary Knott ("O") who always moves second.

Game 1: Mate on the Seventh Stone - combinatorial play

						    a b c d e f g
1)	C1  d1					6   . . . . . . . 
2)	D2  d3					5   . . . . . . . 
3)	D4  b1					4   . . . X . . . 
4)	C2  f1					3   . . . O . . . 
						2   . . X X . . . 
				diagram 1-1	1   . O X O . O .

We label the 42 cells on the board A1-G6 as shown. We have put the 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 stones are about to participate in a quick forced win.

The game continues:

						    a b c d e f g
5)	C3  c4					6   . . . . . . . 
6)	B2  a1					5   . . . . . . . 
7)	A2					4   . . O X . . . 
						3   . . X O . . . 
						2   X X X X . . . 
				diagram 1-2	1   O O X O . O .

Here are two more examples of combinatorial play you may wish to solve for yourself. Eques can force four-in-a-row by her ninth stone in Diagram 1-3 and by her eleventh stone in Diagram 1-4. Unlike the moves above, these games were played in expert fashion, although there were two errors in the moves up to Diagram 1-3.

						    a b c d e f g
1)	D1  e1		5)	C4  f1?		6   . . . . . . . 
2)	A1  c1					5   . . . . . . . 
3)	E2? e3					4   . . X . . . . 
4)	C2  c3					3   . . O . O . . 
						2   . . X . X . . 
				diagram 1-3	1   X . O X O O . 


						    a b c d e f g
1)	D1  d2		5)	G1  e1		6   . . . . . . . 
2)	D3  c1		6)	G2  g3		5   . . . O . . . 
3)	C2  c3		7)	E2  g4		4   . . . X . . O 
4)	D4  d5					3   . . O X . . O 
						2   . . X O X . X 
				diagram 1-4	1   . . O X O . X

Game 2: Second_Player gets Two Stones in the Second Row - positional play

						    a b c d e f g
1)	D1  d2		5)	E3  .. 		6   . . . . . . . 
2)	D3  e1					5   . . . . . . . 
3)	D4  a1					4   . . . X . . . 
4)	A2  e2					3   . . . X X . . 
						2   X . . O O . . 
				diagram 2-1	1   O . . X O . .

Unlike Game 1, where Eques's stones were poised for a quick "combinational" attack based on immediate threats, in diagram 2-1 the game will "drag on" without incident into the ending. But Knott is just as sure of victory as Eques was in diagram 1-1. All he has to do is avoid playing in the first row. Later when X runs out of other moves, X will play in the first row, O will play above her and eventually complete a horizontal 4-in-a-row on the second row. Play the game out a few times until you're convinced that Knott victory is inevitable.

In diagram 2-1, the unoccupied cells C2 and F2 (and B2 and G2 as 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 he will 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 cells and gets what she wants with a sequence (combination) of forcing plays. In Game 2, Knott's win will come in the ending and cannot be hurried.

The "Two Stones in the Second Row" are of value only to the Second_Player. If Eques got this configuration it would be comparatively valueless. In the ending, as the columns are filled in, the First_Player will generally occupy the first, third and fifth rows on a column while the Second_Player gets the second, fourth and sixth rows. Eques therefore wants threats on an Odd Row.

Although in the opening Knott strives for two stones on the second row, it should be noted that the threats can actually be on any even row if there is no odd-row Eques threat underneath them. Furthermore a single stone may be enough if Knott also has a stone on the cell below:

						    a b c d e f g
1)	A1  d1		5)	F3  f4		6   . . . O . X . 
2)	D2  d3		6)	F5  d6		5   . . . O . X . 
3)	F1  f2		7)	F6  ..		4   . . . X . O . 
4)	D4  d5					3   . . . O . X . 
						2   . . . X . O . 
				diagram 2-2	1   X . . O . X .

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 that Eques 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 B4 and C3 and there is no way Eques can ever develop this. With no tactical chances 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 Draw might be possible, but as it stands Knott will win readily with ordinary caution.

The next example (or something very similar) arises constantly in actual play if Knott knows what he's doing and Eques doesn't. Here Eques has no chance for any Odd threats at all and Knott can get seven-in-a-row on the top row if he chooses. An inexperienced Eques player may be pleased with her Major Threats at B2 and F2, but these worthless threats will evaporate under Zugzwang in the ending.

						    a b c d e f g
1)	D1  e1		5)	E4  d6		6   . . . O . . . 
2)	E2  e3		6)	A1  c1		5   . . . O . . . 
3)	D2  d3		7)	C2  ..		4   . . . X X . . 
4)	D4  d5					3   . . . O O . . 
						2   . . X X X . . 
				diagram 2-3	1   X . O X O . .

Let's look now in detail at an expert opening.

Game 3: The 3-4 Opening -- D4 is Poison [Joseki 13]

						    a b c d e f g
1)	D1  d2					6   . . . . . . . 
2)	D3  c1					5   . . . . . . . 
3)	C2  c3					4   . . X . . . . 
4)	C4  ..					3   . . O X . . . 
						2   . . X O . . . 
				diagram 3-1	1   . . O X . . .

Although the rightside is still empty of stones, Eques has great influence there. She is threatening F1 which is quite strong and may build an attack along the C5-G1 diagonal as well. Let us consider the various 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 by E3 or B1 to win easily.

If Knott defends at F1, the variation might be:

						    a b c d e f g
Game 3 variation				6   . . . . . . . 
4)	..  f1 					5   . . O . . . . 
5)	F2  f3					4   . . X X . O . 
6)	D4  c5					3   . . O X . O . 
7)	G1  f4					2   . . X O . X . 
				diagram 3-2	1   . . O X . O X

The Major Threat at E3 will win for Eques (even though Knott has a Major Threat on the same cell). The plays at C5 and G1 could have been reversed. Knott F3 was intended to prevent third row horizontal threats; but D4 would have give Eques somewhat more trouble.

Knott can also put his fourth stone at C5:

						    a b c d e f g
Game 3 variation				6   . . . . . . . 
4)	..  c5		8)	D5  f5		5   . * O X . O . 
5)	G1  f1		9)	A1  a2		4   . . X O . O . 
6)	F2  d4		10)	E1  ..		3   . . O X * X * 
7)	F3  f4					2   O . X O * X . 
				diagram 3-3	1   X . O X X O X

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 minor threat B5 is counted as the equivalent of a Major Threat because of the way threats E2 and E3 work together.) To be sure that Eques can win, we must convince ourselves that the B5 threat will survive any Knott counterplay on the leftside. The only way for Knott to cause trouble is if he gets to occupy both A4 and A5; then the Knott Threat at B4 will neutralize Eques's at B5. Obviously Eques can prevent Knott from getting both A4 and A5 so Eques's threats will survive. (Knott needs A4 since Eques gets a winning Threat at B3 if Eques gets A4).

In other words, Eques's partial diagonal C4-D3 is "undercut" by Knott's partial diagonal C3-D2 but that in turn is "undercut" by the Eques partial diagonal at C2-D1. Finding the bottom-most such diagonal threat is an essential part of most endgame analyses.

In this variation both Eques and Knott played rather well, but there 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 in fact Knott can win after Eques's seventh stone at F3. (Eques should have played D5 first.) Knott failed to win however; he should have played his eighth stone at A1 instead of F5. Even as it was Eques had to play at A1 and 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 D5 to 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 Major Threat to go with her right-side odd threats and is forced to attack on both sides of the board. In the 3-4 opening Eques can play F1, building a Threat at E2 immediately and winning with relative ease:

						    a b c d e f g
Game 3 variation				6   . . . X . . . 
4)	..  d4		8)	D6  ..		5   . . O X . . . 
5)	F1  g1					4   . . X O . . . 
6)	F2  f3					3   . . O X . O . 
7)	D5  c5					2   . . X O . X . 
				diagram 3-4	1   . . O X . X O

In diagram 3-4, Eques will answer G2 with G3, or F4 with F5, and otherwise just take F4. Thus she gets Odd Minor Threats at E5 (and either 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 out to be his toughest defense.

						    a b c d e f g
4)	..  g1					6   . . . . . . . 
5)	G2  g3					5   . . X . . . . 
6)	E1  e2					4   . . X . . . . 
7)	C5  ..					3   . . O X . . O 
						2   . . X O O . X 
				diagram 3-5	1   . . O X X . O

The plays at G2 and G3 correctly block enemy horizontal threats. Eques E1 is a Sente (forcing) move since if Eques plays E2, Eques has a fairly routine endgame win at her B5 Threat cell. Eques C5 was a Gote move (it threatens nothing immediately) but takes the only important central cell which can be taken "for free". (Taking D4 would give the enemy D5 and E3 would give up E4.)

In the game through diagram 3-5, all of Eques's moves have been unconditionally forced (if she wants to guarantee victory) except for her third and fourth stones at C2 and C4. Suppose she decides to play C5 Gote before E1 Sente in the above line. Transposing the Gote stone C5 to be the sixth stone is losing play for Eques:

						    a b c d e f g
Game 3 Variation				6   . . . . . . . 
4)	..  g1					5   . . X X . . . 
5)	G2  g3					4   . . X O . . . 
6)	C5  d4					3   . . O X . . O 
7)	D5  ..					2   . . X O . . X 
				diagram 3-6	1   . . O X . . O

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 by playing A1.

Returning to the main line (see diagram 3-5), what should be Knott's plan? He can't very well take a good cell like D4 or E3 since Eques will get the even better cell above it (D5 or E4). Since he'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 the board locked up with even threats. Of course Eques will not permit this. In fact Eques must catch on to this plan at once (to avoid being forced to take D4 later), and play A1 (or B1) in response to G4 and specifically G4 in response to A1.

In the game, Knott took A1 though the ending might be similar after G4 as well.

						    a b c d e f g
7)	..  a1					6   . . O . . . O 
8)	G4  c6					5   . . X . . . X 
9)	G5  g6					4   . . X . . . X 
10)	E3  ..					3   . . O X X . O 
						2   . . X O O . X 
				diagram 3-7	1   O . O X X . O

Knott doesn't want to open up anywhere so he plays passively on the sixth row. Eques could of course have played E3 the turn before with equal effect.

In the following Eques goes after Major Threats at F5, B3 and B5 so most of Knott's moves are forced.

						    a b c d e f g
10)	..  e4					6   . . O . . . O 
11)	E5  ..					5   . . X * X * X 
						4   . . X * O . X 
						3   . . O X X . O 
						2   . . X O O * X 
				diagram 3-8	1   O . O X X . O

We've interrupted the game briefly to indicate how hard it is for mere human Eques to always win. Wherever Knott now plays, Eques must answer in the same column! (Exceptions: F1 can also be answered with D4. E6 must be answered at A2.)

The Eques threat configuration D4-D5-F5-F2 is a winner, with both Eques and Knott threats in the A column irrelevant.

						    a b c d e f g
11)	..  a2	 				6   O . O . X . O 
12)	A3  a4					5   X . X * X * X 
13)	A5  a6					4   O . X * O . X 
14)	E6  ..					3   X . O X X . O 
						2   O . X O O * X 
				diagram 3-9	1   O . O X X . O

Eques has played passively at E6 (she could also have played there the turn before) to let Knott come to her. Eques has the winning triple-odd threat configuration at B5-D5-F5. Knott's even threats at B2-D4 would defeat Eques if viewed in isolation, but Eques also has even threats at D4-F2 and Knott must now move (ZUGZWANG).

Knott has nothing to gain from letting Eques play F2 and then D5 so the game continues:

						    a b c d e f g
14)	..  b1	 				6   O . O . X . O 
15)	B2  b3					5   X . X * X * X 
						4   O . X * O . X 
						3   X O O X X . O 
						2   O X X O O * X 
				diagram 3-10	1   O O O X X . O

It is now fairly clear that Eques has a simple endgame win. Worth noting 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.

Game 4: The Difficulty of Joseki [Joseki 17]

	    a b c d e f g		    a b c d e f g
	6   . . . . . . .		6   . . . . . . .
	5   . . . . . . .		5   . . O . X . .
	4   . . X . . . .		4   . . X . O . .
	3   . . O . X . .		3   . . O O X . .
	2   . . X . O . .		2   X . X X O . .
	1   X . O X O . .		1   X . O X O . .
	      Diag 4-1			      Diag 4-2


	1) D1 e1 A1 c1 C2 c3 C4 e2 E3 .. 	(Diag 4-1)
	5) .. e4 D2 d3 E5 c5 A2 ..		(Diag 4-2)

In Diagram 4-2 it is apparent that Eques has a routine win. Eventually the play will be B1 b2 B3 and Eques will have a winning threat at D5. Note how the play Eques 8 A2 suddenly brings the stone 2 A1 back to life. This means the stone at A1 had "Aji" (latent value) although it was very latent in this case since A2 would be a premature play for either player any earlier.

The moves through Diagram 4-2 constitute a Joseki: both sides have played 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 Eques got B2 it would establish the cells C3 and D4 as minor threats; since these are central cells the configuration would be very strong for Eques. For one thing Eques could claim D3 since Knott d3 conceding Eques D4 is out of the question.

Alternative Eques plays are 6 E5 and 2 B1 (no other deviations will guarantee victory.) Since Knott is destined to lose in any event against expert 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 to play with care. Note that each of the plays e2-e4-d3-c5 were played where Eques 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 would be 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 all first 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 of Eques's prior play at A1 is relevant.

Since Knott will need to answer an eventual Eques A3 with a4 in any event, the possibility of Eques A1-A2-A3 has no extra value. It appears that the 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 seem superior to 3 C2.

But as we have seen, Eques A1 has a hidden offensive meaning: Eques now has the eventual option of dropping a stone at A2 which works together 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 C2 the only winning play. 3 A2 and 3 G1 lose; other plays Draw. Here is an easy notation to reflect this:

				a2  b1  c2  d2  e2  f1  g1
	X's Response to
		1 D1 e1 A1 c1	O   =   X   =   =   =   O

With the alternative (Eques 3 E2) a position identical to Diag 4-1 may be reached but with the stones at E2 and E3 having color reversed.

	    a b c d e f g		    a b c d e f g
	6   . . . . . . .		6   . . . . . . .
	5   . . . . . . .		5   . . . . . . .
	4   . . X . . . .		4   . . X . X . .
	3   . . O . O . .		3   . . O X O . .
	2   . . X . X . .		2   . . X O X . .
	1   X . O X O . .		1   X O O X O O .
	      Diag 4-3			      Diag 4-4

	1) D1 e1 A1 c1 E2 e3 C2 c3 C4	(Diag 4-3)

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 through Diagram 4-3 also constitute Joseki. To achieve the guaranteed draw, Eques can substitute only 5 D2, 5 E4 or 4 D2 and Knott can substitute only 4 ..d2.

	5) .. d2 D3 f1 E4 b1		(Diag 4-4)

These moves are all logical and required. Knott 5 .. f1 fails because of the combination:

		5) .. f1? D2 f2 B1 b2 D3 d4 E4

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 at b4 and f4, and either of these threats is enough for Knott victory if Eques cannot counterattack. Conversely Knott can force a Draw with relative ease, for example:

		8) G1 c5  F2 d4  D5 e5  B2 f3  F4 f5  F6 b3  B4 b5  B6

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 five potential odd-row threats (A3,B3,D5,F3,G3) and she will have to develop at least three of these to have a chance. Because of the way Knott's diagonals dominate the sides of the board, an Eques threat at A3 is almost worthless without an Eques threat at B3, and G3 similarly depends on F3.

The cell D5 is absolutely essential for Eques (unless Knott carelessly plays b3 or f3 before taking d4 D5). Eques must therefore defend now on the C5-F2-G1 diagonal or Knott will take f2 and either c5 or g1 and force Eques to play D4 d5.

If Eques now takes F2, Knott's C5-G1 diagonal will disappear, while if Eques takes C5 or G1, Knott can take the other cell (Miai cell) and force Eques 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's eventual threat at F3. Instead Eques should take G1 or C5 -- whichever is better. If Knott replies at the Miai cell and forces Eques to take F2, at least Eques will have gotten the better of the two Miai cells.

But C5 and G1 also have minimal offensive value. C5 can contribute only to the A3-B4-C5-D6 diagonal which is worthless unless Eques can undercut the B-column at B3 (ie, occupy A2, A4 or E6). And G1 contributes to no possible four-in-a-row except the unlikely-looking G1-G2-G3-G4.

The game may continue:

	    a b c d e f g		    a b c d e f g
	6   . . . . . . .		6   . . X . O . .
	5   . . X . . . .		5   . . X . X . .
	4   . . X . X . .		4   . . X . X . .
	3   . . O X O . .		3   . . O X O . .
	2   O . X O X X .		2   O O X O X X X
	1   X O O X O O O		1   X O O X O O O
	      Diag 4-5			      Diag 4-6

	8)  C5? g1  F2 a2!	(Diag 4-5)

After Knott makes the fine play of a2, Eques would need E6 to secure the B3 threat. Eques will thus need to build her attack on the right-hand side.

But if she plays something like 10) C6 e5 B2 e6, Knott grabs the remaining 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:

	10) E5 e6  C6 b2  G2	(Diag 4-6)

The exchange C6-b2 can be omitted from this game with a similar final result, but if Eques does play C6, Knott is almost in Zugzwang and the passive play at b2 is his only salvation. Eques needed three odd-threat columns to win 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.

	    a b c d e f g		    a b c d e f g
	6   . . X . O . .		6   . . X . O . .
	5   . . X X X O .		5   . . O X X O .
	4   . . X O X X .		4   . . X O X X .
	3   . . O X O O .		3   . . O X O O .
	2   O O X O X X X		2   O O X O X X X
	1   X O O X O O O		1   X O O X O O X
	      Diag 4-7			      Diag 4-8

	12) .. d4  D5 f3  F4 f5 	(Diag 4-7)

In Diag 4-7 Eques can play 15) D6 g3 A3 a4, but Eques Even Threat B4 is worthless -- it is Knott who will inevitably play at b4 and win in the ending. Eques's attack has simply fizzled out.

Referring way back to Diag 4-4 in which C5 and G1 appeared to be Miai cells, Eques might have played 8 G1 which Knott would answer at c5. If the game then proceeded exactly as above (which it well might since the cells C5 and G1 had minimal effect on the analysis) then a position ALMOST identical to Diagram 4-7 is reached (Diag 4-8). But now Eques wins at once with a double Atari, 15) G3 d6 G4. (After Eques 8 G1, Knott 9 ..a2 is no longer Tesuji but is a subtle blunder; instead Knott must play to draw as mentioned above.)

In the sequence 5) C4 d2 D3 f1 E4 b1 G1, all of the moves are uniquely 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 turn derives its special value only from the tactical possibility of

		8 G1 ... 12 G2 ... 15 G3 ... 16 G4

And this study arose from my failure to see the hidden Aji of Eques A1 in Diagram 4-1. 3 E2 seemed to be a more logical move but I found Eques struggling 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 to memorize Joseki 18. It is very pleasing:

	    a b c d e f g
	6   . . . . . . .	1) D1 e1  A1 e2  B1 c1  B2 b3
	5   . . . . . . .	5) C2 c3  D2 a2  D3 d4
	4   . . . O . . .
	3   . O O X . . .
	2   O X X X O . .
	1   X X O X O . .
	     Diag 4-9

Knott's 2 .. e2 appears logical since it threatens to make the forcing move e3 and also works towards getting Two Stones in the Second Row. Eques can prevail but she must play a precise sequence of forcing moves.

Usually Eques' goal is to get stones in the Third Row, but in this unusual opening she plays in the second row with B2-C2-D2. Knott's plays at b3 and c3 are almost forced (Eques gets very strong threats if permitted to occupy 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 Eques configuration "peeks" out from Knott's blanket at D3 and this is just enough to win.

It turns out that Eques wins easily in this diagram: the threat formation on the rightside (D2-E3-F4-G5, D3-E3-F3-G3, along with the E4 Major Threat) is victorious (it's similar to a "J" but more complicated-looking); Knott's stones on the leftside may look impressive but they can accomplish nothing. The simplest way for Eques to continue is

	    a b c d e f g
	6   . . . . . . .	8) E3 e4  D5 c4  B4
	5   . . . X . . *
	4   . X O O O * .
	3   . O O X X * *
	2   O X X X O . .
	1   X X O X O . .
	     Diag 4-10

Eventually Eques will get F4 and win the ending at G5. If this isn't obvious, play the game out remembering to answer Knott g2 at G3.

Threat Analysis

When an ending is devoid of tactical complications a simple analysis will predict the outcome. This analysis is useful earlier in the game to determine which potential threats are worth pursuing.

Briefly:

Step 1) Analyze the Odd Threats of each player and classify the game as Won for Eques, Won for Knott or Drawn. Even Threats are relevant only when in 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 on an 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 in separate columns.

	    a b c d e f g	    a b c d e f g
	6   . . . . . . .	6   . . . . . . .
	5   . * * X X O .	5   . * X * X * .
	4   . . . . . . .	4   . . . . . . .
	3   * X * X O . .	3   . . . . . . .
	2   . . . . . . .	2   . . . . . . .
	1   . . . . . . .	1   . . . . . . .
	      Eques Wins	      Eques Wins

Just a pair of odd minor threats are enough to win if Eques has an Even Threat in one of the same columns. The Even threat can be a Major threat or it can be half of a Mixed (Odd/Even) Minor threat pair.

	    a b c d e f g	    a b c d e f g           a b c d e f g
	6   . * . . . . .	6   . . . . . . .	6   . . . X . . . 
	5   * * X X . . .	5   . . . . . . .	5   . . * . . . . 
	4   . . . X . . .	4   . . . . X . .	4   . X . O X . . 
	3   . . . . X . .	3   * X * X . . .	3   * O . X . . . 
	2   . . . . . . .	2   . . * . . . .	2   . . * . . . . 
	1   . . . . . . .	1   . X . . . . .	1   . X . . . . . 
	      Eques Wins	      Eques Wins	      Eques Wins

In the absence of Eques threats, Knott needs two Odd Major Threats to win. He also wins with any Even Threat. As seen in Diagram 2-1, the Even Threat can be a Minor threat pair (but not a Mixed Threat), or even a nearly empty row (Diagrams 2-2, 2-3). But Knott's Even Threat is worthless when Eques has winning Odd threats unless the Knott Threat undercuts (is on the same column as and below) the Eques threats.

When both players have Odd Major Threat(s), the important consideration is the total number of columns with Odd Threats. With two such columns, either player needs threats in both of them to win. Eques has the advantage with one or three such columns; Knott has the advantage with four. When both players have two Odd Major Threats Knott wins if the threats occupy two columns, Eques if three, the game is Drawn with the threats on four separate columns.

In counting the odd-threat columns for the above rule, a Knott Minor Threat is counted only if it is a Mixed threat with the Even threat on the same 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 of special value only when adjacent (or a 2nd-row/5th-row pair.) If Knott has Major Threats on the 2nd and 5th rows of the same column, and Eques has an Odd Major Threat elsewhere the game is only Drawn; if a third column contains another Odd Major Threat Knott wins if it is his, the game is still Drawn if it is Eques's.

When one column has threats of each player, Eques dominates when both threats 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 other Even, the lowermost threat dominates.

Here is a configuration which may arise from time to time.

	    a b c d e f g
	6   . . . . . . .	1) D1 d2  D3 d4  D5 e1  E2 e3
	5   . . . X O . .	5) E4 b1  G1 b2  G2 e5
	4   . . . O X . .
	3   . . * X O . .
	2   * O * O X . X
	1   . O . X O . X

Knott will win at A2 eventually, because of his compound Threats in the C-column. But he must be in no hurry to play c2. Instead he must preserve 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-column were of course mistakes.

It is not so easy to analyze early positions but you will often be able to find the best moves if you understand the endgame objectives. It is also easy to go wrong. For example:

	    a b c d e f g
	6   . . . O . . .	1) D1 d2  D3 d4  D5 e1  E2 e3
	5   . O . X . . .	5) E4 b1  B2 b3  B4 b5  G1 d6
	4   . X . O X . .
	3   . O . X O . .
	2   . X . O X . .
	1   . O . X O . X

Both sides have played with skill so far. Eques could have grabbed Odd Minor Threats on the leftside with 8 D6, but victory would still require care 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 Major Threat. If Knott responds at b6 in an effort to Draw with the Counterthreat at c5, Eques can then take 10 F1 which kills ("undercuts") the Knott leftside threat.

But 9 G2 doesn't work. After 9 G2 f1! B6, Eques will discover that the 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 when Knott answers 9 .. g2, 10 C1 is the move to destroy Knott's even Threat at f4.

	    a b c d e f g
	6   . . . O . . .	9) F1 g2  C1
	5   . O . X . . .
	4   . X . O X . .
	3   . O . X O . .
	2   . X . O X . O
	1   . O X X O X X

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 Major Threat eventually, since Eques must get G5 or G6. But if Knott plays 10 .. f2 in the diagram how should Eques respond?

If Eques responds at C2 or E5 to 10 .. f2, Knott will play 11 .. f3 and undercut Eques' rightside threat with the Mixed Minor Threat f4-g5. The Eques partial diagonal D1-E2 is underneath Knott's partial diagonal d2-e3 and this advantage must be preserved. In the diagram Eques must answer f2 at F3.

The following configuration (called the "J") is seen very frequently. Usually it is good enough for Eques victory.

			    a b c d e f g
			6   . . . . . . . 
			5   . . . . . . . 
			4   . . . . . . * 
			3   . . . X X * * 
			2   . . . . X . . 
			1   . . . X . . . 
			 "J Configuration"

When the G-column is eventually played, Eques will get G3 or G4. In either case F3 becomes an Odd Major Threat. Of course it is really the arrangement of the empty (Threat) cells -- shown here with "*"s -- that provides 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:

			    a b c d e f g
			6   . . . . . . . 
			5   . . . . . . . 
			4   . . O O O . . 
			3   . . O X X . . 
			2   . . . . X . . 
			1   . . . X . . . 
			 "Capped J Configuration"

Here Knott intends to play up the F-column since Eques cannot play F3. Eques will get F4, but her "J" is destroyed. Eques can defend by playing up the G-column to convert the F3 Threat into a Major Threat; then Knott cannot play f2 -- or if he has played f1-f2 while Eques takes G1-G2-G3, Knott will be in Double Atari at G4 and F3. Hence in the "Capped J" there will be a race between the players in the F- and G-columns.

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:

			    a b c d e f g
			6   . . . . . . . 
			5   . . . . . . * 
			4   . . . O O * . 
			3   . . . X X * * 
			2   . . . X O . . 
			1   . . O X O . .

(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 both on Odd Rows, they behave like a single Threat.)

The "J" configuration is so common that you should memorize some of the variations that can arise. If Knott plays poorly the "J" will win easily, but in these games Knott plays well.

Example "J" Game 1)

		1) D1 d2 D3 d4 D5 d6 E1 c1 C2 c3 E2 c4 E3 e4 G1 f1 G2

			    a b c d e f g
			6   . . . O . . . 
			5   . . . X . . . 
			4   . . O O O . . 
			3   . . O X X . . 
			2   . . X O X . X 
			1   . . O X X O X

Knott blocks the leftside "J" by playing 5 .. c3, but he would lose quickly 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. As you can see, Eques has won. Here Knott made two questionable moves. 3 .. d6 is usually considered too passive. Having made that play, Knott should answer 4 E1 at f1. Eques could still prevail, but she can no longer force an easy win with "J".

Example "J" Game 2)

		1) D1 d2 D3 d4 D5 e1 B1 a1 E2 b2 E3

			    a b c d e f g
			6   . . . . . . . 
			5   . . . X . . . 
			4   . . . O . . . 
			3   . . . X X . . 
			2   . O . O X . . 
			1   O X . X O . .

Knott could have played 5 .. e3 to stop the "J", but Eques would answer at E4 and, with Even Major Threat C2 already in the bag, would have a very strong attack. (If Knott responds to E4 on the leftside you can win easily on the rightside: 7 G1 g2 G3 g4 G5 g6 E5; while if Knott plays his 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 Threat at c3 and he is threatening to get a Major Threat at c2 as well. Eques was not immediately concerned about this Double Threat, since after 6 .. a2 E4 e5, the Knott c3 threat becomes irrelevant -- Knott would have the leftside under control but no way to stop Eques from winning on the rightside with her "J". The game continues

		6) .. e4 A2 b3 B4 b5 D6 a3 E5

			    a b c d e f g
			6   . . . X . . . 
			5   . O . X X . . 
			4   . X . O O * * 
			3   O O . X X * * 
			2   X O . O X . . 
			1   O X . X O . .

When Knott takes e4, Eques must of course play A2 to prevent the Double Threat. Similarly Knott must then play b3 or Eques will get another Odd 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 that an 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 takes D6 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 demonstrates when she grabs E5. In the diagram, Eques has compound rightside threats (shown with the four "*"s) and will win. Eventually Knott will be forced to play c2, f2 or g2. (Three columns have Odd Threats as required for Eques victory according to the Endgame Principles.)

Example "J" Game 3)

		1) D1 d2 D3 d4 D5 e1 B1 b2 A1 c1 C2 c3 E2 c4

			    a b c d e f g
			6   . . . . . . . 
			5   . . . X . . . 
			4   . . O O . . . 
			3   . . O X . . . 
			2   . O X O X . . 
			1   X X O X O . .

Here Knott tries 4 .. b2, blocks the leftside "J" with 5 .. c3, but succumbs to a rightside "J" due to the Eques Major Threat at E4. Eques can now take her "J" by playing E3 whenever she wants but it is often good enough to grab it at once:

		8) E3 e4 E5 f1

			    a b c d e f g
			6   . . . . . . . 
			5   . . . X X . . 
			4   . . O O O . . 
			3   . . O X X . . 
			2   . O X O X . . 
			1   X X O X O O .

Knott has "capped" the "J" and appears to be winning the race since he gets to drop a stone in the F-column before Eques can drop at G1. But Eques has a trick up her sleeve:

		10) G1 f2 G2 f3 F4

			    a b c d e f g
			6   . . . . . . . 
			5   . . . X X . . 
			4   . . O O O X . 
			3   . . O X X O . 
			2   . O X O X O X 
			1   X X O X O O X

Eques wins since she threatens Eques F5 with a winning Odd Threat at G5, and if Knott blocks at f5, Eques 13 G3 wins at once.

I suggest you memorize these three games in which Eques has played well. Then it will be easy to win with 4 B1 as in Example Games 2 and 3 and you 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. This formation should lead to a Knott win:

						    a b c d e f g
						6   . . . . X . . 
						5   . . X O O * * 
						4   * . O O O * . 
						3   * * X X O . . 
						2   . . X O X . . 
				diagram 5-1	1   X . O X . . .

Eventually the A column will be played and Eques will get A3 or A4. In either case, B3 becomes a Major Threat -- thus B3 can be treated as a Major Threat as it stands. Similarly G5 can eventually be promoted to a Major Threat for Knott. With two odd-threat columns and neither player having threats in both columns, we have an Odd-Threat Draw. Knott will win the game of course since he doesn't actually have to play up the F-column to enjoy the G5 threat: his Even Threat F4 will win the ending.

Usually the Eques partial diagonal at D1-E2 will threaten F3-G4 and will neutralize Knott threat G5. That does not apply here since Eques can hardly play F3. The winning Knott threat configuration is a bit fragile, by the way, and Knott will get only a Draw (or even lose) unless he is careful.

Even without knowing the moves that led to diagram 5-1, a glance suggests that Eques neglected to seize the key cell D5 when she had the chance.

						    a b c d e f g
1)	D1  d2 		5)	C2  e4		6   . . . . . . . 
2)	D3  d4		6)	C3  c4		5   . . . . . . . 
3)	C1  e1					4   . . O O O . . 
4)	E2  e3					3   . . X X O . . 
						2   . . X O X . . 
				diagram 5-2	1   . . X X O . .

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 plurality in the upper center. Knott is even threatening to destroy Eques's "J" by playing up the B column (Eques cannot answer b2 with B3 since Knott wins at b4). To preserve the "J" Eques must drive up the A column immediately:

						    a b c d e f g
7)	A1  b1 					6   . . . . X . . 
8)	A2  e5					5   . . X O O . . 
9)	E6  d5					4   . . O O O . . 
10)	C5  ..					3   . . X X O . . 
						2   X . X O X . . 
				diagram 5-3	1   X O X X O . .

This is the threat configuration shown in Diagram 5-1. Where should Knott 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 then up the F column since F3 and F4 would both be winning cells. But this won't work:

						    a b c d e f g
10)	..  c6	 				6   . . O X X . . 
11)	A3  a4					5   . . X O O . . 
12)	D6  f1					4   O . O O O . . 
13)	B2  ..					3   X . X X O . . 
						2   X X X O X . . 
				diagram 5-4	1   X O X X O O .

Knott gets his Double Threat at F3-F4 but Eques gets a Double Threat at B3-B4 and wins first. In diagram 5-3, Knott can win with any play except C6 or B2.

The tactical complications in diagram 5-3 may obfuscate the essential strategic considerations in diagram 5-2. Suppose that Eques reverses his tenth and eleventh stones:

						    a b c d e f g
7)	A1  b1 					6   . . X O X . . 
8)	A2  e5					5   . . X O O . . 
9)	E6  d5					4   O . O O O . . 
10)	A3  a4					3   X . X X O . . 
11)	C5  d6					2   X . X O X . . 
12)	C6			diagram 5-5	1   X O X X O . .

Knott must now take care to grab the critical G-column cells before playing even a single stone in the F-column:

		12) .. g1 G2 a5 G3 a6 G4 g5

Eventually Knott will play f1 and there will be no stopping his Double 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 have other winning sequences besides those shown in diagrams 5-3 and 5-5. But the key configuration shown in diagram 5-1 is ultimately Knott's target.

Problems

	 			Problem A	(Joseki 14)
				---------
	 . . . . . . . 		1. D1  d2
	 . . . O . . . 		2. D3  c1	("X" = Eques = 1st player)
	 . . . X . . . 		3. C2  c3	("O" = Knott = 2nd player)
	 . . O X . . . 		4. D4  d5
	 . . X O . . . 
	 . . O X . . . 		Eques to play and win.


	 			Problem B	(Joseki 2)
				---------
	 . . . X . . . 		1. D1  d2	7. B4  b5
	 . O . X . . . 		2. D3  d4	8. D6  g1
	 . X . O X . . 		3. D5  e1
	 . O . X O . . 		4. E2  e3	Eques to play and win.
	 . X . O X . . 		5. E4  b1
	 . O . X O . O 		6. B2  b3


	 			Problem C	(Joseki 4)
				---------
	 . . . . . . . 		1. D1  d2	7. B3  b4
	 . X . X O . . 		2. D3  d4	8. G1  g2
	 . O . O X . O 		3. D5  e1	9. B5  c1
	 . X . X O . X 		4. E2  e3      10. G3  g4
	 . X . O X . O 		5. E4  b1
	 . O O X O . X 		6. B2  e5	Eques to play and win.


	 			Problem D	(Joseki 13, Game 3)
				---------
	 . . . . . . . 		1. D1  d2	Eques to play and win.
	 . . . . . . . 		2. D3  c1
	 . . X . . . . 		3. C2  c3	After Eques makes her best
	 . . O X . . O 		4. C4  g1	play,  Knott answers at A1.
	 . . X O O . X 		5. G2  g3	Where should Eques now play?
	 . . O X X . O 		6. E1  e2


	 			Problem E	(Game 4)
				---------
	 . . . . . . . 		1. D1  e1	7. E4  b1
	 . . . . . . . 		2. A1  c1
	 . . X . X . . 		3. E2  e3	Eques to play with what result?
	 . . O X O . . 		4. C2  c3
	 . . X O X . . 		5. C4  d2
	 X O O X O O . 		6. D3  f1


	 			Problem F	(Joseki 19)
				---------
	 . . . . . . . 		1. D1  e1
	 . . . . . . . 		2. B1  a1
	 . X . . . . . 		3. B2  b3	Eques to play and win
	 . O . . . . . 		4. B4  a2
	 O X . . . . .
	 O X . X O . .

Solutions

Problem A

Eques must play G1 and go after the D4-G1 diagonal. If she tries to force the issue with

	5.  F1 c4  G1 e1  F2 c5  C6 b1

she loses to Knott's counterattack on the upper left side. When Eques does play 5 G1, a pretty continuation is

	5.  G1 e1  G2 g3  E2 g4		(Diag 1-4)

with Eques then able to force four-in-a-row by her tenth stone. Here Eques must 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 Major Threat at F2 appears to neutralize the rightside, but Eques intends to play C4 eventually and thereby kill the F2 threat. But first things first! Eques must start at B6 since if Knott plays there, Eques C4 is no longer playable. In some variations Eques will eventually win at B6-C6-D6-E6.

Problem C

Eques can play up the A-column and then the C-column with a series of Sente moves, eventually winning on the C6-D5-E4-F3 diagonal. But Knott has a Major Threat at F4 and can therefore foil Eques's plan by playing F1-F2-F3. It's a race and Eques wins if she plays A1 immediately.

Problem D

The third-row cells are generally the best cells on the board. Good players avoid playing on the second row (especially at D2) if it gives their opponent the better cell above it. But fourth-row and fifth-row cells are also valuable. Thus here the great value of E4 cuts down on the value of E3. D4 is almost unplayable for either player 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 Eques should play there. It is a "free" play: C6 has little value.

On the topic of "avoiding the second row", consider the common opening

	1.  D1 c1  C2 ...

in which Knott and Eques respectively did and didn't follow this advice. The strongest continuation is then

	2. ... c3  C4 g1

Both Victor and I wasted many computer-hours on this position in a fruitless search for Eques victory: the game is drawn.

Problem D2

After Eques C5 in Problem D, Knott takes A1. Knott hopes to get G4 followed by D4 which would "lock up" both sides of the board with even-row threats. If Eques lets Knott get G4, Eques will have to then 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 with his b4 and f4 threats unless Eques can build threats at B3 and/or F3. Hence Eques wants to occupy cells like A2, A4, E6, etc but especially D5. But she cannot afford to play A2 now since then she loses the key cell D5:

	8.  A2 f2  C5 g1  D4 d5

Thus Eques must play along the C5-G1 diagonal now to have any chance. None of these cells has much offensive value but, hard as it is to foresee, the possibility of G1-G2-G3-G4 makes G1 the correct play and if Eques plays anywhere 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 will establish Odd Minor Threats at A5 and C3 and the main virtue of A3 is to protect 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 are no prime cells available. As usual the preeminent value of D3 makes D2 a poor Eques play despite the diagonal connection with B4 (5 D2 d3 A3 e2 is very strong for Knott.) E2 is a good-looking play for Eques since it works towards an Even Threat at C2 or C4. But against 5 E2 Knott will not waste a tempo taking a5, as he would against 5 B5. Instead he will take e3 and (with b3 in the bag already) Eques would no longer have Third Row Horizontal possibilities. (After E2 an expert continuation might be

	5   E2 a3 A4 e3 D2 a5 B5 d3 G1 g2 G3 g4 E4 e5 E6 b6 A6

with Eques finally blocking on the sixth row just to salvage a Draw.) After Eques makes the correct play of A3, one example continuation is

	5   A3 e2 E3 e4 A4 a5 E5 b5 D2 d3 D4

Further Notes on the Problems

Problem F

Note the bizarre leftside-heavy position after 5 B4 a3 A4 a5, even though the entire opening (except 5 B4) is expert. One good way to see the answer is to first realize that A3 and E2 are the most valuable available cells, but that E2 "gives up" more -- E3 is also very valuable, but A4 isn't. The opening position (empty board) is won for Eques, but 3 of the 4 distinct 1-stone positions are won for Knott. Among 25 distinct 2-stone positions, 14 are won by Eques, 5 by Knott and 6 are drawn.

Among 121 distinct 3-stone positions, 18 are won by Eques, 74 are won by Knott and 29 are Drawn. Details of this are shown next.

First Move Results

			Legend
			------

		"X" --> X (Eques, First Player) Wins after the indicated Move
	        "O" --> O (Knott, Second Player) Wins after the indicated Move
	        "=" --> Drawn Game after the indicated Move

				 - - - -  Move  - - - -
				a   b   c   d   e   f   g
				-------------------------

	X's First Move:		O   O   =   X   =   O   O

	O's Response to
		1 A1		X   O   O   O   X   O   X
		1 B1		X   =   O   =   X   X   X
		1 C1		X   X   =   =   =   =   X
		1 D1		X   X   X   X   X   X   X

	X's Response to

		1 A1 a2		O   =   =   X   O   X   O
		1 A1 b1		O   O   O   O   O   O   O
		1 A1 c1		O   O   O   O   O   O   O
		1 A1 d1		O   O   O   O   O   O   O
		1 A1 e1		O   O   O   X   O   O   O
		1 A1 f1		O   O   O   O   O   O   O
		1 A1 g1		O   O   O   X   O   O   O

		1 B1 a1		=   X   =   O   X   O   O
		1 B1 b2		O   O   O   O   =   O   O
		1 B1 c1		O   O   O   O   O   O   O
		1 B1 d1		O   =   O   =   O   O   O
		1 B1 e1		O   O   O   X   O   O   O
		1 B1 f1		O   X   =   X   X   =   O
		1 B1 g1		O   X   =   X   =   O   O

		1 C1 a1		=   =   =   X   =   =   O
		1 C1 b1		O   =   O   O   =   X   O
		1 C1 c2 	O   =   O   =   O   O   O
		1 C1 d1		O   O   =   =   O   O   O
		1 C1 e1 	O   O   =   O   =   O   O
		1 C1 f1		O   =   =   =   =   =   O
		1 C1 g1		O   =   X   X   =   X   O

		1 D1 a1		=   O   X   X   X   X   X
		1 D1 b1		O   X   O   =   =   X   O    [Joseki 26]
		1 D1 c1		O   O   =   =   O   X   X    [Joseki 16 - 25]
		1 D1 d2		O   O   O   X   O   O   O

	X's Response to

		1 D1 d2 D3 b1	O   X   O   X   X   X   O
		1 D1 d2 D3 c1	O   O   X   X   O   X   X    [Joseki 13 - 15]
		1 D1 d2 D3 d4	O   O   O   X   O   O   O    [Joseki  1 - 12]

Results assume that all remaining moves are played as "joseki" (best way for both players). For example if Eques starts at D1 and Knott answers a1, Knott can force victory if Eques plays B1, the game is drawn with best play if Eques plays A2, and Eques can win with any other play.

There isn't room to show Knott's second stone in the lines where Eques has blundered; the best play is often on the third row or in the D column. Of course there are many exceptions: In the variation

	C1 f1 F2 c2 C3 c4

Knott 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 page as "Joseki."

Opening Game Tree ("Joseki")

		(Variations with 1 .. d2 or 1 .. b1 )

   x-1-o x-2-o x-3-o x-4-o --5-- --6-- --7-- --8-- --9-- -10-- -11-- -12-- 

   D1 d2 D3 d4 D5 e1 E2 e3 E4 b1 B2 b3 B4 b5 G1 d6 F1 g2 C1 f2 F3 f4 F5    [1]
    |     |     |  |           |  |  \ E5  | 
    |     |     |  |           |  |        \ D6 g1 B6                      [2]
    |     |     |  |           |  |           \ f1 G1
    |     |     |  |           |  |              \ C1 f2 G1 f3 F4 e5 etc.  [3]
    |     |     |  |           |  |   
    |     |     |  |           |  \ e5 B3 b4 G1 g2 B5 c1 G3 g4 A1 f1 etc.  [4]
    |     |     |  |           |     |           \ A1  \ A1
    |     |     |  |           |     \ G1        \ C1
    |     |     |  |           |     |
    |     |     |  |           |     \ D6 g1 E6 b3 B4 b5 B6 f1 F2 f3 F4    [5]
    |     |     |  |           |
    |     |     |  |           \ E5 b2 B3 a1 A2 a3 A4                      [6]
    |     |     |  | 
    |     |     |  \ B1 e2 A1 c1 C2 c3 C4 b2 C5                            [7]
    |     |     |     | 
    |     |     |     \ a1 E2 b2 E3 e4 A2 b3 B4 b5 D6 a3 E5                [8]
    |     |     |        \ B2  \ A2 b3 B4
    |     |     |
    |     |     \ b1 B2 b3 B4 f1 F2 f3 F4 b5 F5                            [9]
    |     |     |           | 
    |     |     |           \ b5 F1 e1 E2 d6 E3                           [10]
    |     |     |                       | 
    |     |     |                       \ e3 G1 d6 E4 f2 F3 g2 C1 f4 F5   [11]
    |     |     |                          \ D6
    |     |     \ d6 E1
    |     |        \ C1 b1 F1 e1 E2 e3 E4 b2 C2 e5 C3 c4 C5 g1 F2 b3 etc. [12]
    |     |
    |     \ c1 C2 c3 C4 g1 G2 g3 E1 e2 C5 a1 G4 c6 G5 g6 E3 e4 E5 a2 etc. [13]
    |        |     |                             \ E3
    |        \ F1  \ D4 d5 G1 c4 B1 c5 C6 e1 E2                           [14]
    |        \ G1           |
    |        \ D4           \ e1 G2 g3 E2 g4                              [15]
    |
    \ b1 F1 e1 F2 f3 D2 d3 C1 d4 F4                                       [26]
       \ B2              \ D4
                         \ F4

						(continuations)
   				     -12-- -13-- -14-- -15-- -16-- -17--

			 [3]	     C2 f5 C3 c4 C5
			 [4]	     A2 a3 A4 c2 C3 c4 C5 a5 C6 f2 F3
			[12]	     B4 b5 F3
			[13]	     A3 a4 A5 a6 E6 b1 B2
			            \E6   \E6


		(Variations with 1 .. e1 )


   x-1-o x-2-o x-3-o x-4-o --5-- --6-- --7-- --8-- --9-- -10-- -11-- -12-- 

   D1 e1 A1 d2 D3 d4 B1 c1 C2 c3 E2 c4 D5 c5 C6 f1 D6                     [20]
       |  |  \ B1              \ C4
       |  |                    \ D5
       |  |
       |  \ c1 C2 c3 C4 e2 E3 d2 E4 e5 F1 d3 D4 d5 F2                     [16]
       |                    |        \ D3
       |                    \ e4 E5
       |                       \ D2 d3 E5 c5 A2                           [17]
       |
       \ B1 a1 B2 b3 B4 a2 A3 d2 D3
          |        \ D2     \ e2 E3 d2 D3 d4 D5 d6 B5                     [19]
          |                       \ b5 D2 d3 E4
          |                       |        \ D4
          |                       |        \ A4
          |                       |
          |                       \ e4 E5 b5 D2 d3 A4 d4 A5               [21]
          |                          |  |        \ D4
          |                          |  |
          |                          |  \ d2 B5 g1 G2 c1 D3
          |                          |     \ D3        \ C2
          |                          |
          |                          \ A4 a5 E5 b5 D2 d3 D4
          |
          \ e2 B2 b3 D2 e3 E4 a1 D3 d4 B4 a2 D5 a3 A4 a5 E5               [22]
          |  \ E3  \ A1        \ B4  \ G1  \ G1        \ C1 c2 E5 b5 G1
          |  |     \ C1        \ E5  \ D5  \ E5        \ G1
          |  |     \ E3
          |  |
          |  \ A1 c1 B2 b3 C2 c3 D2 a2 D3 d4                              [18]
          |
          \ b2 A1 c1 C2 c3 C4 e2 C5 a2 A3 b3 B4 d2 D3
                   |     |        |  \ E3        \ E3
                   |     |        \ g1 G2 b3 B4 e3 E4                     [23]
                   |     |        \ c6 A2 or E3
                   |     |
                   |     \ E2 c4 C5 d2 D3 d4 D5 d6 G1
                   |           |           \ G1 d5 A2 f1 A3 a4 G2 f2 G3   [24]
                   |           |                       \ G2
                   |           |
                   |           \ G1 c5 C6 a2 G2
                   |                       \ E3 e4 E5 e6 G2 d2 G3 g4 D3   [25]
                   |                                   \ A3  \ D3
                   |
                   \ E2 b3 B4 a2 C2
                      |     \ c2 A2
                      |     \ d2 D3
                      |     \ e3 B3
                      |
                      \ c2 A2 b3 B4

Another important Joseki begins with the moves

		1 D1 e1 B1 b2

This opening is played so strongly by both players that we shall consider it in some detail. 1 .. e1 is probably Knott's best chance, since otherwise Eques can win by memorizing the key Joseki [7] - [12]. Similarly the play 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 at the key central cell D4 is worth far more than the diagonal pointing at B4. In fact Eques must seize A1 at once. For example in Joseki [23]:

		3 A1 c1 C2 c3 C4 e2 C5

				. . . . . . .
				. . X . . . .
				. . X . . . .
				. . O . . . .
				. O X . O . .
				X X O X O . .

(Eques can substitute E2 as her 5th or 4th stone.) Since Knott must not let Eques get E4, Eques effectively has E3 and E5 "in the bag" already. Although the rightside of the board is barren, Eques has established a 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 in Zugzwang 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 a1 with Knott winning.

Another variation is Joseki [25]

		3 A1 c1 C2 c3 E2 c4 G1 c5 C6 a2 E3 e4 E5 e6 G2 b3 B4

				. . X . O . .
				. . O . X . .
				. X O . O . .
				. O O . X . .
				O O X . X . X
				X X O X O . X

(Eques can substitute 10 A3, 8 G2, 6 C5, 5 C4 or 4 E2.) Since this is such a stout defense by Knott, eleven stones are not enough for Eques to establish an obvious triple threat, but her victory is inevitable, despite Knott's counterthreat at d5, due to Zugzwang. Knott cannot try f1 and let Eques get a quick win with the forcing moves D2-G3-D3-F3. Eventually Knott will play at d2 for lack of anything better but his punishment won't stop: if he takes d4 he loses his counterthreat and temporizing at f1 fails because of Eques's tactical threat along the D4-G1 diagonal.

Eques can also win this opening by playing her fourth stone at E2 instead of C2. This may be an interesting study. The Joseki chart above 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 Opening Game Tree. Strong Knott moves have been chosen, although space considerations permit only a very small sample of the possibilities. But given the chosen Knott defense every possible winning Eques move is shown. This is rather remarkable and means that Eques just barely has a won game.

Joseki [12] is the "most forced" game in Connect-Four since Eques's first thirteen plays are all uniquely forced. Even if Eques is content to Draw, 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 play 5) B1? e4 and give up the rightside. I spent several hours manually confirming that Eques 9 B6 is required in the very frequently encountered Joseki 2; this study prompted the design of the search software.

In Joseki 4 (another "5-4" Opening) Eques wins with a spectacular endgame race: she must play A1 before Knott plays f1. There are many other Knott defenses possible in the "5-4" Opening. After practicing this opening to gain an appreciation of Eques's and Knott's objectives, you will do well to 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 configurations as shown in the diagrams. In Joseki 7, Knott takes Two Stones on the Second Row, 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 cells instead of three. Eventually Knott will be in Zugzwang, forced to allow Eques to play F3 or G3 with a winning double-threat or C3 with an easy win.

	.X*XX..       .......       ...X...       .......       .......
	.O*XO..	      ..XX**.       .O.XX..       .O*X*X*       ......*
	.X.OXX.	      ..XO*..       .X.OO**       .X.O.X.       ...O**.
	*O.XOO.	      ..OX...       OO*XX**       .O.X.O.       .OOX***
	.X.OXX.	      .OXOO..       XO.OX..       .X.O.X.       OXXXO..
	.O.XOOO	      XXOXO..       OX.XO..       .O.X.O.       XXOXO..
      (Joseki 5)    (Joseki 7)    (Joseki 8)    (Joseki 9)    (Joseki 18)

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 play may 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 to see 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-g3 diagonal -- but she gets "Super-sente" and in the end wins with a rare triple atari!

Joseki 13 (a "3-4" Opening) is also played with utter precision and is rather subtle. Neither player wants to take D4 and give up D5 to the enemy. After 7) C5, Eques's stones dominate the board and with an eventual stone at E5, Eques gets a win similar to that shown in Joseki 6.

Joseki 15 is interesting because Eques never actually establishes a winning endgame; instead she uses "Super-sente" to execute a rare combination and in the diagram can play E3 and announce "mate in two" with her two double-atari threats.

In Joseki 16 - 26, Knott disdains an early d2. In Joseki 18, Eques can hold off Knott's plurality in the upper left and will eventually win on the 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 for position in the corners before it is known who will get the commanding central 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 and John Tromp, another Connect-Four programmer. The author played first in 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, openings which the computer search had revealed to result in Draws with optimal play.

The match was tied: 3 wins to John, 3 wins to the author and no Draws. To conform with our usual convention the First and Second players will be called Eques and Knott.

Here are the moves of the games up to the resignations. The second line shows what the result would have been with perfect play after each move. The third line gives the optimal play(s) when an error was made (including the opening "errors" agreed to).

	 --1-- --2-- --3-- --4-- --5-- --6-- --7-- --8-- --9-- -10--
Game 1
	 D1 c1 G1 d2 D3 d4 F1 e1 E2 e3 C2 e4 D5 e5 E6 b1 D6
	 +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  

Game 2
	 D1 e1 B1 e2 B2 b3 D2 e3 E4 a1 D3 d4 B4 a2 D5 a3 A4 a5 E5
	 +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  

Game 3
	 B1 b2 E1 d1 D2 d3 D4 d5 D6 e2 E3 f1
	 -  =  =  =  =  =  =  =  =  =  -  -
	(D1 c1)                        F1

Game 4
	 C1 b1 B2 b3 D1 d2 B4 e1 E2 d3 D4 c2 C3 a1 D5 d6 B5 g1 B6
	 =  +  =  =  -  -  -  =  -  -  -  +  +  +  +  +  +  +  +
	(D1 d1 F1)   E1       d3 D3       d5
	          or B4

Game 5
	 B1 b2 E1 d1 E2 e3 D2 e4 E5 a1 G1 d3 D4 b3 B4 g2
	 -  =  =  =  -  =  =  =  =  =  -  -  -  -  -  -
	(D1 c1)      D2 b3             B3

Game 6
	 C1 b1 B2 f1 B3 b4 F2 f3 F4 e1 F5
	 =  +  =  +  -  +  +  +  +  +  +
	(D1 d1 F1)b3 F2 f2

Comments on the Games.

Game 1

was another of the many variations where Eques constructs a "J configuration" but Knott "caps it" with a fourth-row horizontal threat. Knott plays 8 .. b1 in an effort to take b3 before Eques can take A3; this is the usual defense to the "J." But Eques finds the only winning response at D6: if Knott takes b3 later, Eques can grab B4 and get odd threats on the A3-C5 diagonal.

Game 2

demonstrates that 2 .. e2 is a weak play in this opening; Eques cannot be allowed to build up the leftside unmolested. With 7 B4, Eques establishes a threat configuration at A5-C3-C2. Knott destroys this threat by pushing up the A-column, but while he's spending tempi doing this Eques seizes D5 and E5. The Eques stones at D3-D5-E4-E5 are equivalent to a "J" and give Eques a win at F5. Knott resigns: he could take g1 and undercut the F5 threat but Eques would then simply take B5.

Game 3

became drawish when neither player got two vertically adjacent stones. Eques of course was wrong to allow 6 .. f1: a diagonals rooted on the first row can never be undercut.

In the opening diagonals which point towards central cells like D4, D3 or even D2 are more valuable than the peripheral diagonals which D1 controls. Thus in

Game 4

Eques 3 D1 was a mistake, and B4 or E1 was preferred. (Knott can answer 3 F1 with e1 and win.) In this game each player was at first reluctant to take D3 -- it seemed like D4 was the essential cell. But post-game analysis revealed that this position was no exception to the rule that D3 is the best cell (he who gets D3 may get D5 as well) and both 4 .. e1 and 5 E2 were errors.

An interesting position developed after 6 D4. Knott was afraid that Eques would play A1 and establish an Odd Major Threat so 6 .. c2 looked right -- it defended the leftside and established an Even Major Threat on the rightside. But, although Eques could no longer establish an ordinary-looking "triple threat", her huge plurality in the upper leftside meant she could win easily as long she dropped a stone at 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. Now Knott will presumably get e3 and e5 eventually and the equivalent of a "J configuration" on the rightside: he already has the Odd Major Threat C3 and either f3 or f4 gives him a Major Threat at G3.

In Game 5

Eques was unhappy with her play 4 D2 since Knott then took adjacent stones at e3-e4, but a Draw could still have been salvaged. After 5 E5 the players exchanged weak-looking moves in the corners, but a1 was a subtle and cunning play, while G1 was a blunder.

This was a game the author lost, but with hindsight the best plays should be clear with a little study. C1 and F1 are worse than useless for either player. E6 is also almost worthless: it doesn't even work towards a Knott Even Threat at C4 because of b3-B4. G1 is also of minimal value. B3 and D3 are the "prime real estate" available but they give up B4 and D4 which are even better. After the exchange b3-B4, B3-b4, d3-D4 or D3-d4 there will be strong plays available, so each player wants to time these plays for best effect.

If Knott answers 5 E5 with d3-D4-d5 it will be fairly clear to Eques to play to Draw. 5 .. b3 is also bad since B4 is such a better cell. By process of elimination Knott should play a1 as he did; note that he's threatening to follow with b3 and establish a certain Major Threat at C3. He's also threatening a2, after which either B3-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 drawish result. Perhaps she was thinking to pick up G1-G2 with some rightside strength, but she should know from her own studies that she cannot survive such a dreadful "double Gote" sequence.

Knott announced victory immediately when Eques blundered and made the precise forcing moves d3-D4-b3-B4-g2. (Here only d5 for b3 and a2 for d3 can be substituted.) Knott gets another Major Threat at F3, but doesn't really need it: the triple threat c2-c3-f5 might be good enough. Eques would have liked to have grabbed G2 for herself, but after 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 not 4 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 grabs d3, but so will Knott and that is the important thing. Knott failed to take advantage: he should grab b3 in response to 3 E2, and if Eques then takes B4, grab d2.

In Game 6

Knott resigned after playing only five stones. He has little influence on either the 2nd or 3rd rows. Eventually the play will be d2-D3-e3-E4-c2-C3 with Eques getting an Odd Major Threat no matter how Knott plays. The computer search to find Knott's error is not complete. My guess is that 2 .. b3 was appropriate: Eques is always very happy to get the first stone on the third row.

Consider the line

	1 C1 b1 B2 f1 B3 f2 B4 b5 F3 g1 G2 c2 C3 e1 D1 e2 D2 d3 B4 e3 D4 e4

Almost all of the results given have been established with exhaustive computer search. There are two exceptions, where a simple manual analysis will suffice.

--- Knott wins after 1 A1 ------

Adopt the following strategy: Answer A1 at d1, A2 at a3, B1 or B3 at b2 or b4, C1 at c2, E1 at e2, F1 or F3 at f2 or f4. Answer any other move in the D-column. But if D6 is already occupied answer at F1 or F5 if possible, or in the B-column if c2 is occupied; F3 works if c2 is not occupied.

Following these rules you will eventually place a stone at an even-row cell in the D-column or at an odd-row cell in the B- or F-column. At that point you will have a relatively straightforward win and should "switch off the autopilot."

Another way to look at this strategy is that Eques will eventually take her choice of A4, B5, C3, D6, E3, F5 or G1 and then you strike back. Of these D6 and F5 are Eques's most promising plays. If she takes F5 and gives you d6, she will follow with F6 in an effort to establish a threat:

			...O.X.
			...O.X.
			.?.X.O.
			??.O.X.
			???X?O.
			X??O?X.

Here Knott will eventually win at A6-B6-C6-D6. (Uncontested Even Threats are so good for Knott that he prevails easily even though his threats do not fulfill the definition of even "minor" threats.) The only possible Eques counterthreat is on the diagonal C3-D4-E5-F6. But if C1 c2 have already been 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 of these columns instead.

Eques may decide to take D6 and give you f3:

			...X...
			...O...
			.?.X...
			??.O.O.
			??.X?O.
			X?.O?X.

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 neutralized since Knott answers F4 at c1, or vice versa. If Eques had played C1 c2 before D6, Knott would answer in the B-column and not at f3.

--- Knott draws after 1 C1 ------

Victor Allis established with a simple argument that Knott can guarantee at least a Draw after 1. C1 d1. (He has a similar argument for 1. A1 d1 and 1. B1 c1.) Knott simply plays above each Eques stone, playing in the C-column when Eques plays D6 or in the D-column when Eques plays C6. Sooner or later Eques must take D2-D4-D6 to block. The forced configuration is thus

			OO?XOOO
			XX?OXXX
			OO?XOOO
			XX?OXXX
			OO?XOOO
			XXXOXXX

which is obviously a draw.

To Draw when Eques opens at B1, answer B1 at c1, F1 at g1 or vice versa; play at any even-row cell when Eques plays in the sixth row and otherwise 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 the autopilot."

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.

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