# Drawing squares

Anafenza and Zurgo are playing a game. They have a playing board of $23*92$ squares.

The idea of the game is to put a square(of any size) onto the playing board. The one who can't put any more squares loses the game.

Zurgo is hastening, so he gets the priority and puts his first square on the board. Then it's Anafenza's turn.

Can Zurgo guarantee himself a victory? Can Anazenza guarantee herself a victory? Or will the game end in a draw?

As always, we assume that Zurgo and Anafenza use optimal strategies.

• Isn't this the same as Chomp? – Rand al'Thor Mar 17 '15 at 11:08
• @randal'thor nope, not really – Novarg Mar 17 '15 at 11:11

If Zurgo places a 22x22 square on the board so that there are two areas of 23x35 and one of 1x22 left, he has created a board on which he can mimic every move by Anafenza.

That means that no matter what move Anafenza can make, Zurgo can copy, which means that Zurgo will always be able to move - unless Anafenza just lost!

Starting with a smaller square is tricky - the point is to divide the board into two mirrored halves. a 1x1 square will not leave equal areas in the lenth direction of the borad (46x23 versus 45 by 43) - this is true for any odd-numbered starting square!

For smaller even-sized squares, the odd number of squares left "next" to it makes it again impossible to mimic. If Zurgo plays a 20x20 square, Anafenza can play a 3x3 square next to it (in the remaining 3x20 area) that can not exactly be mirrored.

(in the 1x20 area Anafenza can only play a 1x1 square which can always be mirrored in the same area)

Zurgo, who goes first, should place a square in the middle and then mimic Anafenza on the opposite side of the board.

In the event Zurgo does not place a square in the middle, Anafenza should then mimic Zurgo.

-Edit-

This strategy works for one simple reason if Zurgo is mimicking Anafenza this means that there must be an equivalent spot on the opposite side of the board, of which Anafenza played, for a square to be drawn.

• Welcome to Puzzling.SE! Answers here should generally come with explanations - could you explain why this strategy is optimal? – Rand al'Thor Mar 17 '15 at 11:12
• A 23x23 square in the center of the board doesn't leave an even space on both sides - one side will have a 34x23 space, while the other has a 35x23 space. – mdc32 Mar 17 '15 at 12:28
• @mdc32: But a 22x22 square does leave 35x23 on each side, plus a 22x1 space that can also be divided. So indeed, Zurgo can indeed mimick Anafenza. – oerkelens Mar 17 '15 at 14:12
• I should have been more clear I intended the initial square to be a 1x1 – user2097159 Mar 17 '15 at 15:00
• With a 1x1 square, your tactic doesn't work... – oerkelens Mar 17 '15 at 15:10

Zurgo

because the Game started by the Zurgo. my simple logic is

23x92 = 2116

2116/2 = 1058

1058/2 = 529

529/2 = 264.5 which is having 0.5 fraction value. it means clearly it will end with Zurgo and there will no option to place a Squre for Anafenza