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April - May 2010, Puzzle 124
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 A chopping problem
Using the checkered pattern as a guide, cut out the board below into 7 rectangular pieces so that no piece can contain another one (for instance, a 6x8 rectangle completely covers a 4x7 rectangle).

Difficulty level: bulbbulb, basic geometry knowledge.
Category: dividing-the-plane puzzle.
Keywords: dissection, tiling, rectangles.
Related puzzles:
- Squared strip,
- Triangles to square.

Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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So, each of the 7 rectangles should have one side WIDER and one side SHORTER to each other rectangle, so that neither of the rectangles can be placed inside the other one in such a way that corresponding sides are parallel.
Since we have to find 7 rectangles that tile a 22 x 13 units2 rectangle, we will proceed empirically as follows:
Rectangle 1: 1 x ... units2
Rectangle 2: 2 x ... units2
Rectangle 3: 3 x ... units2
Rectangle 4: 4 x ... units2
Rectangle 5: 5 x ... units2
Rectangle 6: 6 x ... units2
Rectangle 7: 7 x ... units2

Rectangle 1 should have the widest surface since it has the shortest height; however, the possible dimensions 1 x 22 and 1 x 21 should be discarted [any rectangle with sides n ≤ 13 x (22 - 21) would fit inside the latter one]... Then, proceeding by trial and error we obtain the following tiling:

1 x 18
2 x 16
3 x 13
4 x 11
5 x 10
6 x 9
7 x 7
(see image below)

solution 1

Aside from rotations and reflections this tiling of rectangles is unique

cup winnerThe 5 Winners of the Puzzle of the Month are:
Herbert Jones, USA USA flag - Charles F. Espenlaub, USA USA flag - Cesco Reale, Italy Italian flag - Muhammad Afifi, Egypt Egyptian flag - Martin Rick, South Africa South African flag


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Math fact behind the puzzle
Incomparable Rectangles
Two rectangles, neither of which will fit inside the other, are said to be 'incomparable' (this is equivalent to one rectangle being both longer and narrower). The minimum possible number of incomparable rectangles needed to tile a larger rectangle is 7


© 2006 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
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