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. | ||
| + | [/list] | ||
| + | [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 06:19, 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.
[/list]
[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 
is it possible that, after some non-zero number of moves, the board has no stones?
