## Dots and Squares

In the classic game, two players alternate turns, adding a line between two dots.  In a turn, if a player creates a unit square, they claim that square for their score and must then add another line (if two squares are created at once, they still add only a single line.)  When a player places a line and no square is created, the next player takes a turn.  In each puzzle below, it is Player 1's turn.

 1. The game has reached the state to the left.  Only one line will guarantee Player 1 the win.  Which one?
 2. The game has reached the state to the left.  Player 1 can guarantee a win with what strategy?
 3. The game has reached the state to the left.  Player 2 can guarantee a win with what strategy?

Source: Original.

Solutions were received from Alan O'Donnell, Kirk Bresniker, Yaacov Yoseph Weiss, Dan Chirica, and David Stignant.  Below is a summary of the answers, followed by Yaacov's summary of the solutions.

For #1 and #2, player 1 will lead to a win by playing the green line of the 2-square section.  If he plays either end of the 2-square section, player 2 could lead to a win by playing the opposite end.  For #3, player 2 can win by splitting the square when player 1 plays there (play to complete the green or red line.)

Yaacov's summary:

The key nonobvious strategy is that when completing a block, you should often leave the last 2 squares for the other player, forcing him to offer another block. A corollary is that a block of 2 should be offered in such a way that the opponent must take, and cannot offer it back. (It is a controlling option, if it is better for opponent to take, he will take anyway.)

1: Thus in the first case, the key is to offer the pair, without allowing a counteroffer. Then give 2 squares from the block the opponent chooses, and take the whole complete block. Player1 wins 5-4.  [KD: Yaacov also pointed out that this problem can be solved through symmetry - the highlighted line is the only unique good move.]

2: The single blocks in the corner are traded, and then the key is (as before) offer the pair on the top right without allowing a counteroffer. Opponent gets one of single squares, the pair, and 2 pairs from the 3 blocks. Total of 7/16. player 1 wins. Note that player 1 can start by offering a singleton. Any response from 2 gets the same response it would get later, except that if p2 offers the pair, (even allowing a counteroffer,) p1 takes and offers the second single.

3: Again the singles in the corners get traded. The idea is the same also here. If p1 first offers the 5s, p2 gets 1+3+3+4=11.  If p1 offers the 4, p2 is forced to not take it at all, leaving those 4 for p1. p2 compensates with taking a complete set of 5 later, totalling 1+3+5=9.
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