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Our gaming articles: Tic-tac-toe by Silverman.

# Gaming article "Tic-tac-toe by Silverman"

(With our redaction of rules.)

Tic-tac-toe is a pretty famous game within America. But at the same time, it is usually only played by small children due to its simplicity and easy mechanics - which is sometimes a problem for adults.

For example, tic-tac-toe can easily be played into a tie game using simple strategies; for instance when one player places a token in the corner, his/her opponent should place their token in the center, and vice-versa. Another problem with the typical tic-tac-toe game is the size of the game board: 3x3 - which is much too small. In this article we shall consider the variant offered by Silverman: a revised version with a 4x4 grid for added complexity. The new version also had turn-base rules. If a row of four identical tokens (either X's or O's) appears somewhere on the board - horizontally, vertically or on a diagonal - the player who started the game wins. Otherwise the second player wins. In this version the first player (the player who goes first) has the greatest opportunity to win if he/she can correctly position the tokens before their pattern is blocked by the opponent.

The individuality of this revised version comes through the strategies given to the first player. At the start of the game, Player 1 "attacks", threatening to create a row of four tokens, then diverges from the plan by making neutral moves to give Player 2 an opportunity to create a four-in-a-row line of O's. We can see that in this updated version of the game there can be no draw.

Computer-based calculations show that in this new version, the first player (X tokens) has the winning strategy. But, for the second player (O tokens) to win, he/she must only use generic reasoning to find the best position for their token - to block the opportunity for Player 1 to line up his/her tokens. For example, the first player may put his/her first token in a corner; influencing the second player to put his/her token in the opposite corner to block the diagonal line-up.

Therefore we shall consider another variant of the game in which the advantage of token placement will be weakened by such rule: If four identical tokens appear on the top-right to bottom-left diagonal, it is not a winning position, and the play continues. (This is our correction of the original rules.) In this case, the game proceeds as usual. If such a variant of the game becomes interesting enough, both players should often make some unique and unforeseen moves.

Fact #1: All of the first moves of Player 1 (X tokens) to the right diagonal (from the top-right to the bottom-left diagonal) loose and all the rest win. To make everything clear we shall use chess notation to coordinate the gaming area, along with your moves.
Fig. 1 shows all of the positions where Player 2 (O tokens) can win by blocking the first player's (X tokens) move to the corner:

Fig. 1.
Let's suppose that O's have countered Player 1's move to the top-left corner with the following move:
1. b4
Then consider the possible variants of the game after this move. Let X's move be displayed in Fig. 2:
2. d1

Fig. 2.
The second player in this position can only win with move in Fig. 3:
2. ... a1
This move is obvious as it blocks possible vertical and horizontal fours of X's.
But X's continue to attack:
3. c2
O's again have one and only one move:
3. ... b3

Fig. 3.
In this position X's make a fork with a 2x2 move shown in Fig. 4:
4. d2
O's again have one and only one move:
4. ... a2
(If the second player answers 4. d4, then the first will win with move 5. a2.)

Fig. 4.
In this position the first player tries to compel the second player to position his/her O's in places that would turn out to benefit Player 1 (Fig. 5):
5. d4 d3
6. c1

Fig. 5.
Here is the critical moment of this game. The second player needs to block the vertical four-in-a-row opportunity Player 1 has in the column C.
If player two moves to c4, he/she wins, and Player 1 loses.

Now lets consider a game where the first player (X tokens) moves somewhere other than the main diagonal, show in Fig. 6:

1. b4 c4

Fig. 6.
Now, what moves do you think Player 1 should make to win in this situation,
2. b3 (Fig. 7)?
In fact, this move threatens column b, the third horizontal and the main diagonal (advantage to Player 1). Absolutely not! In this position, an O can win by the only one move
2. ... d3!
However, we will continue this game after Player 1's move to
2. b3?

Fig. 7.
At first, it seems as if Player 2 has two equivalent moves:
2. ... b1 or
2. ... b2
But in this situation, the winning move proves to be to
2. ... b1! (Fig. 8).
Let's watch what happens. X's continue to attack:
3. a3 a4
(3. ... c3 will also result in a win here)

Fig. 8.
We continue in Fig. 9:
4. d3 c3
5. d2 d4

Fig. 9.
X's are still threatening the O's with row 2. However, there is a good chance the O's won't loose their winning positioning this game. (Fig. 10):
6. a2 b2

Fig. 10.

It becomes clear that if Player 1 (X tokens) moves to
7. a1,
then Player 2 (O tokens) will be followed with
7. ... d1
and vice versa. In this situation the Player 1 will lose.

Now it becomes very interesting to learn why in the original game, Player 2's move to
2. ... b2
is a losing one. Considering an O was placed in
2. ... b2,
Player 1 would have two great counter-moves to make:
3. d1
and
3. d3
Let's consider these moves (Fig. 11):
2. ... b2
3. d1

Fig. 11.
The second player then answers (Fig. 12):
3. ... a4,
As he not wants to make a two-by-two fork at c2.
The first player continues:
4. d3
Which becomes the only winning move

Fig. 12.
O's obviously answer with (Fig. 13):
4. ... d4
Then X's move:
5. b1
Which strengthens his/her position. The second player is compelled to put another O in a line, that contains only O's:
5. ... a1

Fig. 13.
X's have the last attacking move, shown in Fig. 14:
6. c3
And this time O's have to make a row in a vertical as an answer to this attack:
6. ... a3
Compare this with Fig.10 where they have made the sixth move without loosening their position.

Fig. 14.

With this move, X's attack comes to an end can only make "wait-and-see" moves:
6. c1
6. c2 or
6. d2
Their position is so strong that all these moves lead X's to the victory in this game.

Now, let's consider the game after three moves (Fig. 15):
1. b4 c4
2. d4

Fig. 15.
Here we can see that O's have six different moves which block the verticals b and d; but every move from this one has a winning continuation move for the X's. Let O's play in the position
2. ... d3
Then the X's follow (Fig. 16):
3. b2!
(We shall consider later why the move 3. b1? move is a loosing one.)
3. ... b1

Fig. 16.
The first player has to attack, otherwise he/she will loose (Fig. 17):
4. a2
The second player obviously protects him/(her)self
4. ... d2
X's attack develops easily:
5. a4
(Fig. 17.)

Fig. 17.
And the second player puts another O on a line where there is already one (Fig. 18):
5. ... a1
X's still have positions of the main diagonal to use for their attack:
6. b3

Fig. 18.
Here O's are better to protect themselves with (Fig. 19):
6. ... c2
X's answer with a wait-and-see move:
7. a3

Fig. 19.

But O's do not have wait-and-see moves. In this situation, O's lose.

And now let's consider why Player 1's move to b1 in the original game would be disadvantageous:
3. b1?

Fig. 20.
In this situation, O's make a winning move (Fig. 21):
3. ... b2
4. a1

Fig. 21.
In this position the second player can make two winning moves:
4. a4
or
4. d1
Let's play these (Fig. 22):
4. a4
Then we can watch the following development of the game:
5. c1 d1

Fig. 22.
X's need to block the O's possibility of a diagonal-line and horizontal-line win. In this situation, Player 2 has "wait-and-see" moves for the same lines. For example (Fig. 23):
6. a2 d2

Fig. 23.
But X's loose because of the absence of "wait-and-see" moves.

This game is currently available in two variants:
- for mobile devices with Java 2 ME:
http://www.iqflash.com/d/ttt.jar
- and as a flash game.

At the end I offer you several problems for your practice. The player whose turn is to move wins the game with this very move.

Fig. 24.

Fig. 25.

Fig. 26.

Fig. 27.

Fig. 28.

Fig. 29.

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