Difference between revisions of "2004 IMO Problems/Problem 3"

 
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== Problem ==
 
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]
+
<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]
+
</asy>
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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.
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==Solution==
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{{solution}}
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==See Also==
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{{IMO box|year=2004|num-b=2|num-a=4}}

Latest revision as of 23:53, 18 November 2023

Problem

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 $m \times n$ 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.

Solution

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See Also

2004 IMO (Problems) • Resources
Preceded by
Problem 2
1 2 3 4 5 6 Followed by
Problem 4
All IMO Problems and Solutions