Difference between revisions of "2004 IMO Problems/Problem 3"
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| − | Define a | + | 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] | ||
| + | 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, | ||
| + | (b) no part of a hook covers area outside the rectangle. | ||
Revision as of 22:29, 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]
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.
