Difference between revisions of "2004 IMO Problems/Problem 3"
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Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. | Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. | ||
| − | + | <asy> | |
unitsize(0.5 cm); | unitsize(0.5 cm); | ||
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draw((2,1)--(2,3)); | draw((2,1)--(2,3)); | ||
draw((3,1)--(3,3)); | draw((3,1)--(3,3)); | ||
| − | + | </asy> | |
| + | |||
Determine all <math>m \times n</math> rectangles that can be covered without gaps and without overlaps with hooks such that; | Determine all <math>m \times n</math> rectangles that can be covered without gaps and without overlaps with hooks such that; | ||
(a) the rectangle is covered without gaps and without overlaps, | (a) the rectangle is covered without gaps and without overlaps, | ||
(b) no part of a hook covers area outside the rectangle. | (b) no part of a hook covers area outside the rectangle. | ||
Revision as of 23:32, 20 February 2021
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
![[asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy]](http://latex.artofproblemsolving.com/e/f/4/ef4cdb21baa20a696547d85c9c5773637ad93506.png)
Determine all 
rectangles that can be covered without gaps and without overlaps with hooks such that;
(a) the rectangle is covered without gaps and without overlaps,
(b) no part of a hook covers area outside the rectangle.
