Difference between revisions of "2021 USAMO Problems/Problem 3"
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− | + | Let <math>n \geq 2</math> be an integer. An <math>n \times n</math> board is initially empty. Each minute, you may perform one of three moves: | |
+ | [list] | ||
+ | [*] If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. | ||
+ | [*] If all cells in a column have a stone, you may remove all stones from that column. | ||
+ | [*] If all cells in a row have a stone, you may remove all stones from that row. | ||
+ | <asy> | ||
+ | unitsize(20); | ||
+ | draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); | ||
+ | fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); | ||
+ | draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); | ||
+ | draw((0,2)--(4,2)); | ||
+ | draw((2,4)--(2,0)); | ||
+ | </asy> | ||
+ | For which <math>n</math> is it possible that, after some non-zero number of moves, the board has no stones? |
Revision as of 07:21, 18 July 2021
Let be an integer. An board is initially empty. Each minute, you may perform one of three moves: [list] [*] If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. [*] If all cells in a column have a stone, you may remove all stones from that column. [*] If all cells in a row have a stone, you may remove all stones from that row. For which is it possible that, after some non-zero number of moves, the board has no stones?