Difference between revisions of "2021 USAMO Problems/Problem 3"
| Line 4: | Line 4: | ||
[*] 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. | ||
| − | < | + | <asy> |
unitsize(20); | unitsize(20); | ||
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); | draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); | ||
| Line 11: | Line 11: | ||
draw((0,2)--(4,2)); | draw((0,2)--(4,2)); | ||
draw((2,4)--(2,0)); | 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? | For which <math>n</math> is it possible that, after some non-zero number of moves, the board has no stones? | ||
Revision as of 06:21, 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.
![[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]](http://latex.artofproblemsolving.com/0/7/1/071cfa01f4b3baffd591a2e465d1f3ecfbc58046.png)
For which 
is it possible that, after some non-zero number of moves, the board has no stones?
