Strategy

After learning short-term (in) tactics, a player is equipped to consider long-range planning, or strategy. A player with the knowledge of strategy can often predict the outcome of a game many moves in advance.

Even and Odd Numbers of Spaces

After White moves 16. C6 in this game, Black decides to apply some strategic thought to see if White will be able to win with his threat at G5.

Position after 16. C6
	A	B	C	D	E	F	G
6								Move List
5								1. D1 D2 2. D3 D4 3. D5 D6 4. E1 F1 5. B1 C1 6. C2 C3 7. C4 C5 8. E2 F2 9. E3 E4 10. E5 F3 11. F4 B2 12. A1 B3 13. F5 F6 14. B4 B5 15. B6 E6 16. C6 A2 17. A3 A4 18. A5 A6 19. G1 G2 20. G3 G4 21. G5
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For White to get G5, Black must first move G4. Both players will avoid playing G4, Black to avoid losing, and White to prevent Black from taking G5 and achieving a draw. How many spaces can be played in before G4 is the only option left? There are the remaining spaces in the A-column, and G1, G2, and G3. 5+3=8, an even number. After an even number of pieces are placed on the board, it will be Black's turn again (8 pieces means 4 Black-White pairs). Therefore, Black must eventually play G4 and lose. Whenever the player on turn finds that there are an even number of spaces before a certain location on the board which each player wants the other to take, it means that the player on turn will be the one to move in that location (unless there are tactical complications).

In the next game, after White moves 13. F6, Black tries to see if White will be able to win at C4.

Position after 13. F6
	A	B	C	D	E	F	G
6								Move List
5								1. D1 D2 2. D3 D4 3. D5 D6 4. F1 E1 5. E2 E3 6. E4 E5 7. E6 B1 8. B2 B3 9. B4 B5 10. B6 F2 11. G1 F3 12. F4 F5 13. F6 G2 14. G3 G4 15. G5 G6 16. A1 A2 17. A3 A4 18. A5 A6 19. C1 C2 20. C3 C4 21. C5 C6
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How many spaces can be played in before C3 is the only option left? 6+5+2=13, an odd number. White will not win at C4; this game is a draw.

Even and Odd Boards

Counting spaces can be made easier by using the concept of even and odd boards. To get started, consider an empty 7x6 board:

The game begins
	A	B	C	D	E	F	G
6								Move List
5
4
3
2
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The 7x6 board can be termed an even board since there are 7x6=42 spaces (an even number) on it. As the game has not yet begun, it is obviously White's turn. On an even board, whenever there are an even number of spaces remaining to be filled, White is to move.

On an odd board, like the 7x7, it is White's turn whenever an odd number of spaces are empty.

The game begins
	A	B	C	D	E	F	G
7								Move List
6
5
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Knowledge of even and odd boards is straightforward to apply. Consider again the game where Black wonders if White can win at C4:

Position after 13. F6
	A	B	C	D	E	F	G
6								Move List
5								1. D1 D2 2. D3 D4 3. D5 D6 4. F1 E1 5. E2 E3 6. E4 E5 7. E6 B1 8. B2 B3 9. B4 B5 10. B6 F2 11. G1 F3 12. F4 F5 13. F6 G2 14. G3 G4 15. G5 G6 16. A1 A2 17. A3 A4 18. A5 A6 19. C1 C2 20. C3 C4 21. C5 C6
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Rather than adding the numbers of empty spaces in the different columns, Black could realize that when every possible move other than C3 has been played, there will be 4 spaces left on the board (C3, C4, C5, and C6). Since 4 is an even number, and the game is on an even board, at that point it will be White's turn. White will play C3 and Black will block the threat at C4.

The ability to make calculations this way leads to some new ways of classifying threats.

Even-above and Odd-above Threats

An even-above threat is a threat which, like White's at C4 in the last game, is below an even number of spaces. An odd-above threat, of course, is below an odd number of spaces, like Black's one at E3 in the next game:

Position after 16. G5
	A	B	C	D	E	F	G
6								Move List
5								1. D1 D2 2. C1 B1 3. E1 F1 4. C2 C3 5. C4 D3 6. D4 D5 7. D6 C5 8. C6 G1 9. A1 A2 10. A3 A4 11. A5 A6 12. F2 F3 13. G2 F4 14. F5 F6 15. G3 G4 16. G5 G6 17. B2 B3 18. B4 B5 19. B6 E2 20. E3 E4 21. E5 E6
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As this is an odd-above threat, when it is about to become playable the number of empty spaces on the board must be an odd number (the spaces above the threat, the threat space itself, and the space directly below it). The game has an even board, so it is Black's turn when this happens, and he must give up the threat.

One Column Remaining

It can be seen that if an even board is (or will be) completely filled except for one column (or part of one), White will then occupy any odd-above threats in this last column, and Black any even-above ones. On an odd board, the opposite happens. Consider this game:

Position after 27. ... G9
	A	B	C	D	E	F	G
9								Move List
8								1. C1 C2 2. C3 C4 3. F1 E1 4. E2 C5 5. E3 B1 6. E4 E5 7. E6 A1 8. C6 C7 9. C8 C9 10. F2 F3 11. F4 F5 12. B2 B3 13. A2 A3 14. A4 E7 15. E8 E9 16. F6 F7 17. F8 F9 18. B4 B5 19. A5 B6 20. G1 B7 21. B8 B9 22. G2 A6 23. A7 A8 24. A9 G3 25. G4 G5 26. G6 G7 27. G8 G9 28. D1 D2 29. D3 D4 30. D5 D6 31. D7 D8
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The D-column is full of threats, but all of White's are odd-above and therefore useless. Of Black's, only the odd-above threat at D8 is useful, the rest of his being even-above.

This table summarizes the result of different threats in the last remaining column:

Threat	Even Board Result	Odd Board Result
White odd-above	White wins	Draw
White even-above	Draw	White wins
Black odd-above	Draw	Black wins
Black even-above	Black wins	Draw

Two Columns Remaining

Position after 15. ... A4
	A	B	C	D	E	F	G
6								Move List
5								1. D1 D2 2. D3 D4 3. D5 D6 4. E1 F1 5. B1 C1 6. C2 C3 7. E2 C4 8. E3 E4 9. E5 C5 10. C6 E6 11. G1 G2 12. G3 G4 13. G5 G6 14. A1 A2 15. A3 A4 16. A5 A6 17. B2 B3 18. B4 B5 19. B6 F2 20. F3
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On this even board, White has an odd-above threat at F3, while Black has an even-above threat at B4. Which player wins? After each player exhausts all the move possibilities that do not involve giving up his/her threat or surrendering to the opponent's, an odd number of spaces remain on the board: B3-B6 and F2-F6. Therefore, Black is to move, and must either lose at once by playing F2, or sacrifice his/her own threat. If Black does give up B4, the game simplifies to a one-column situation, in which Black still loses to White's odd-above threat.

However, if the threats of this last game had been on an odd board, the game would have been a draw. Analysis of positions in which two empty columns remain yields the following predictions:

Column 1 Threat	Column 2 Threat	Even Board Result	Odd Board Result
White odd-above	Black even-above	White wins	Draw
White odd-above	Black odd-above	Draw	Draw
White even-above	Black odd-above	Draw	Black wins
White even-above	Black even-above	Black wins	White wins
White even-above	White even-above	Draw	White wins
White odd-above	White odd-above	White wins	White wins
Black even-above	Black even-above	Black wins	Draw
Black odd-above	Black odd-above	Black wins	Black wins