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# INHALT

11 April 2010
Some thoughts about the game 4 in a row
Preamble:
In the evening of some vacation in France we played 4 in a row.
After I recognized that there exist some mathematic rules to win the game in the long run I wanted to write this down.
Please feel free to leave any comment for me being able to improve this short description.

All following rules apply independent of the total number of columns and rows of the game. We assume that white starts the game by placing the first chip and black continues.

Picture 1 describes a situation were black has already won the game provided that:
1) Black prevents white to set up 4 in a row
2) Black does not throw in a chip in column F until white placed F1
3) Black does not throw in a chip in column G until white placed G1

Since there are 32 chips left and it is whites turn at the moment, white will be forced to place chips at F1 and G1 after column A to E are filled up completely.

This text is about how such a situation can be arranged and what needs to be considered without difficult calculations.

Description:
1A) Black is able to force the opponent to throw in a chip, if the number of empty positions in the column is even.
1B) This works for several columns since the result of an even number added to an even number is an even number as well.

2A) White is able to force the opponent to throw in a chip, if the number of empty positions in the column is odd.
2B) But remember when two odd numbers are added to each other the result is an even number. Therefore white has the disadvantage to be able to force black only in 1 column. But it is still possible for white to setup more than one trap and simply not use them. The last trap which is left open by white will force black to throw in his chip.
2C) If an odd number and an even number are added to each other the result is odd. Therefore if both - black and white - have setup a trap, white is able to force black to disable his own trap and will win this match with his trap (still open).

1) If we would like to place a chip at a position 1 above the "skyline" (another chip needs to be thrown in before the chip can be placed at the wanted position) but the opponent could force us to throw our chip first and place his chip above (at the position we wanted our chip to be at), we need to throw in our chip immediatelly to clarify if the other wants to block this. How likely it is that the opponent overlooks something is nothing we want to calculate therefore we want to clarify this immediatelly.
2) Since we are no computers it seems to be a good idea to start with a tower in column D (D1 - D6). This makes it easier to see were the opponent could win with 4 in a row. In the situation of the following figure black has only the chance to win horizontal or diagonal by using D3 or D5. Except those two locations we only need to consider attempts to establishe vertical towers of 4 in a row. (referring to picture 2)

Countermeasurements:
If white would place a chip in D1 and C2 the trap is setup. This is because white can force black to throw in D2, to win diagonal with D3. Therefore to prevent the opponent using this technique, keep an eye 3 positions above the skyline. When D3 is identified as a potentional dangerous scenario black is able to prevent this by placing a chip at C2. (referring to picture 3)

Summary:
Although it might seem to be much easier to play with black (even + even => even), according to the rules described above white has the pretty big advantage to be able to force black in case both have setup their traps. Therefore in my opinion it is more desirable to play white. Actually I have no idea how black could win this game considering both are playing without making a mistake.
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