2021 USAMO Problems/Problem 3: Difference between revisions
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[*] If all cells in a column have a stone, you may remove all stones from that column. | [*] 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. | [*] If all cells in a row have a stone, you may remove all stones from that row. | ||
\begin{asy} | <cmath>\begin{asy} | ||
[asy] | [asy] | ||
unitsize(20); | unitsize(20); | ||
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draw((0,2)--(4,2)); | draw((0,2)--(4,2)); | ||
draw((2,4)--(2,0)); | draw((2,4)--(2,0)); | ||
\end{asy} | \end{asy}</cmath> | ||
For which <math>n</math> is it possible that, after some non-zero number of moves, the board has no stones? | 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.
\[\begin{asy}
[asy]
unitsize(20);
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0));
fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey);
draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2));
draw((0,2)--(4,2));
draw((2,4)--(2,0));
\end{asy}\] (Error compiling LaTeX. Unknown error_msg)
For which 
is it possible that, after some non-zero number of moves, the board has no stones?
