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Tiling Trial (Posted on 2014-07-24) Difficulty: 4 of 5
Is it possible to find tiling of a square into an odd number of non-rectangular pieces each having identical shapes and the same area? (Regard a given piece as identical to another if the rotation and/or reflection of the first piece is identical to the second)

If so, provide an example. If not, prove that it canít be done.

No Solution Yet Submitted by K Sengupta    
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Solution re: Solution Comment 2 of 2 |
(In reply to Solution by broll)

This tiling sounds tough to lay out at first.  But it is not that hard with a good strategy.


Start with the L triomino and find any rectangle with odd dimensions it can tile. In this case 15 tile a 5x9 rectangle:
AABBCCDDE
AFFBGCDEE
HHFGGIIJJ
HKLMMNIOJ
KKLLMNNOO
Then use the trivial tiling of 45 of these rectangles into a large 45x45 square, using 675 total L triominos

This process can also be applied to some of the pentominos.  The L and P pentominos each tile a 7x15 rectangle, so they will each tile a 105x105 square using 2205 pentominos.

The Y pentomino tiles a 15x15 square, and is likely the smallest square that satisfies the problem.

Edited on October 26, 2016, 10:50 am
  Posted by Brian Smith on 2016-10-26 10:50:03

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