n-in-a-row Games, Part I

Some years back, I challenged a friend to a long (100-game) series to determine who would be the World Champion Tic-Tac-Toe player. What started as a jest developed into a serious interest into the world of n-in-a-row games. In these posts, I over-analyze simple things, show the depth of the better games, and try to instill an interest in them.

Tic-Tac-Toe

, three in a row...

Tic-Tac-Toe is proven to be a draw with perfect play, but it is a good starting point to understand the principles of n-in-a-row games.

As everyone in the western world knows, in a regular game of Tic-Tac-Toe, there are nine squares, in three rows and columns. Each player, X and O, take turns placing one of their pieces on an unoccupied square. If there is three-in-a-row of one piece, that player wins. Otherwise the game is a draw (or "cat" game).

Breaking it down more mathematically, we see that on a three-by-three (3x3) Tic-Tac-Toe board, there are eight possible rows of three: the two diagonals, three horizontal, and three vertical. Because the board layout is symmetrical, each corner and edge square initially have identical value. This also means that there are only three openings: center, corner, and edge. In general, a square that has more possible n-in-a-rows through it is a more powerful square, so the center square, which can be in four possible rows, would seem to be the strongest starting position.

	a	b	c
3
2		X
1

In this situation, O has two options, a corner or an edge. After subtracting the rows that are made unwinnable due to the first player's move, any corner has two rows that go through it; any edge has only one, so the corner is mathematically and intuitively the stronger square.

(If we play the edge, we find that the edge defense loses; X plays either a corner or an edge diagonal to the edge chosen, forcing O to defend against 3r, then X can immediately set up a 3r threat from both of his previous squares.

Center Game, Edge Blunder, Corner Variation
1. b2 a2
2. a1+ c3
3. c1#

	a	b	c
3		O
2	O	X
1		X	X

On the other hand, after the corner move, although X is in the stronger position and has the initiative, he is usually unable to win. Most moves he makes will set up a 3r threat, but can also be immediately and directly countered by O.

Center Game, Corner Line
1. b2 a1
2. a3+ c1+
3. b1+ b3
4. c2+ a2~ (draw)

	a	b	c
3	X	O
2	O	X	X
1	O	X	O

X does have one possible option for a trap:

Center Game, Trap Sprung
1. b2 a1
2. c3 a2+
3. a3#

	a	b	c
3	X		X
2	O	X
1	O

The proper defense is to play on a corner.

Center Game, Trap Avoided
1. b2 a1
2. c3 c1+
3. b1+ b3 ... and draw

	a	b	c
3		O	X
2		X
1	O	X	O

This sums up the center game.

---

Because of the simplicity of the defense against it, the center opening is sometimes eschewed in favor of a corner attack.

	a	b	c
3
2
1	X

Consider the position O is in. A corner is taken, meaning that there are only five out of the eight possible rows that O can still win. The center now has three such paths; each remaining corner has two. The edges orthogonal to the occupied corner have one, but the others still have two.

The corner and edge defenses are fatally flawed in a similar fashion to the edge defense previously discussed.

Corner Game, Opposite Corner Blunder
1. a1 c3
2. c1+ c2
3. a3#

	a	b	c
3	X		O
2
1	X	O	X

Corner Game, Adjacent Corner Blunder
1. a1 c1
2. c3 b2
3. a3#

	a	b	c
3	X		X
2		O
1	X		O

Corner Game, Opposite Edge Blunder
1. a1 b3
2. a3+ b2
3. c1#

	a	b	c
3	X	O
2	O
1	X		X

Corner Game, Adjacent Edge Blunder
1. a1 a2
2. b2+ a3
3. c1#

	a	b	c
3			O
2	O	X
1	X		X

---

So, O moves in the center.

	a	b	c
3
2		O
1	X

Because of O's move, X's square is now worse than it was previously, and is actually worth a little less than O's. Having the first move advantage means he keeps the momentum, but this is only enough to guarantee a draw. Why, then, does anyone prefer the corner?

If O isn't careful, she may wander into a trap.

The favorite X response is normally to play the corner opposite to his initial move, which sets up the corner trap: if O moves in one of the other corners, X takes the third corner, blocking 3r and creating two 3r threats of his own, winning the game.

Corner Game, Corner Trap Sprung
1. a1 b2
2. c3 a3+
3. c1#

	a	b	c
3	O		X
2		O
1	X		X

A more subtle but similar variation is the edge trap. X plays in one of the edges adjacent to the opposite corner. If O plays in the wrong corner (the corner furthest from the other two squares), X can again block her 3r attempt and create two threats of his own.

Center Game, Edge Trap Sprung
1. a1 b2
2. b3 c1+
3. a3#

	a	b	c
3	X	X
2		O
1	X		O

However, both of these traps require specific wrong moves from O to succeed. With other easily found moves, the game will end with a draw after a series of blocked 3r attempts. Some games may run out of possible rows before they run out of moves.

Corner Game, Corner Trap Avoided
1. a1 b2
2. c3 a2+
3. c2+ c1+
4. a3+ b3+
5. b1 and draw

	a	b	c
3	X	O	X
2	O	O	X
1	X	X	O

Corner Game, Edge Trap Avoided
1. a1 b2
2. b3 a3+
3. c1+ b1 and an easy draw

	a	b	c
3	O	X
2		O
1	X	O	X

---

Some players have criticized the edge opening and claimed that it loses outright. This is false, but it does require more precise play from X than the other lines. However, because some players are unfamiliar with this system, this may be a way to try for a win.

There are unsound defenses from O that lose by force. These are the adjacent edge and opposing corner.

Edge Game, Adjacent Edge Blunder
1. a2 b3
2. a3+ a1
3. b2#

	a	b	c
3	X	O
2	X	X
1	O

Edge Game, Opposite Corner Blunder
1. a2 c1
2. a1+ a3+
3. b2#

	a	b	c
3	O
2	X	X
1	X		O

On the other hand, the center, opposing edge, and adjacent corners are all solid defensive options, with good chances for O if X makes a mistake.

Edge Game, Center Defense
1. a2 b2
2. b3 a3+
3. a1+ a3+
4. c1+ b1 and draw

	a	b	c
3	O	X	O
2	X	O
1	X	O	X

Edge Game, Symmetrical Defense
1. a2 c2
2. c1 a1 and draw

	a	b	c
3
2	X		O
1	O		X

Edge Game, Adjacent Corner Defense, Opposite Blunder
1. a2 a1
2. c2+ b2+
3. c3+ c1#

	a	b	c
3			X
2	X	O	X
1	O		O

---

There are a couple of principles that we can observe in this. The first is that the direct approach seems less successful in an n-r game. The attacks with the best chance of working involve delayed actions that produce multiple threats. Anyone can notice when you have two pieces lined up in a row against him, but it is possible to miss the point of a 'developing' move and respond with a play in a bad square.

The second is that when it comes to balanced games, you don't win so much as wait for your opponent to lose. If your adversary intends to draw, and moves with the purpose of blocking you as much as possible, there's no reason they shouldn't be able get what they're after. They are in danger when they decide to play a tricky game and try to 'steal' a victory. Of course, the larger and more complicated the game, the less applicable this is.

On that note, stay tuned for the next part, where we discuss some more advanced n-in-a-row games - Tic-Tac-Toe on larger boards, both balanced and imbalanced games.