## Dissecting to Five Rectangles

1. Find the smallest square which can be dissected into five non-overlapping rectangles with integer sides, such that the ten dimensions of the five rectangles are all unique. That is, the length and width of each piece is different from the length or width of any other piece.
2. Find the smallest area rectangle which meets these requirements.
3. Extra Credit. The smallest rectangle above is MxN, but one of the smaller rectangles has N as one of its dimensions. Can you find the smallest area MxN rectangle such that M and N are different from the 10 dimensions of its divided rectangles.
Extension: One reader asked: "What's the next fewest number of pieces?" For example, could six smaller rectangles be found, all with different dimensions? Seven?

Source: Denis Borris and Joseph DeVincentis. Problem 2 was published by Frank Rubin in the Journal of Recreational Math under the name "Temple of Heterodoxy" a few years ago.

Solutions were received from Al Zimmermann, Philippe Fondanaiche, Alessandro Fogliati, Adrian Atanasiu, Alan O'Donnell, Jeremy Galvagni, Jimmy Chng Gim Hong, Luke Pebody, Claudio, Baiocchi, Gary Mulkey, Radu Ionescu, Paul Botham.
Al Zimmermann's text solutions to the first problems are below. Some nice HTML files were created for better graphics and can be found here:
Five Rectangles from Claudio Baiocchi, and More Rectangles from Philippe Fondanaiche.
```The smallest squares are 11 x 11:

A A A A A A B B B B B       A A A A A A B B B B B
A A A A A A B B B B B       D D E E E E B B B B B
A A A A A A B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D E E E E B B B B B
D D E E E E B B B B B       D D C C C C C C C C C
D D E E E E B B B B B       D D C C C C C C C C C
D D C C C C C C C C C       D D C C C C C C C C C

The smallest rectangles are 13 x 9:

A A A A A A A A A B B B B       A A A A A A A A A B B B B
D D E E E E E E E B B B B       A A A A A A A A A B B B B
D D E E E E E E E B B B B       A A A A A A A A A B B B B
D D E E E E E E E B B B B       D D E E E E E E E B B B B
D D E E E E E E E B B B B       D D E E E E E E E B B B B
D D E E E E E E E B B B B       D D E E E E E E E B B B B
D D C C C C C C C C C C C       D D E E E E E E E B B B B
D D C C C C C C C C C C C       D D E E E E E E E B B B B
D D C C C C C C C C C C C       D D C C C C C C C C C C C

The smallest rectangles the lengths of whose sides are distinct from
the lengths of the sides of its dividing rectangles are the two
11 x 11 squares shown above.  However, if we require the rectangle
to be non-square, then the smallest such rectangles are 16 x 8:

A A A A A A A A A A A A B B B B       A A A A A A A A A A A A B B B B
D D D E E E E E E E E E B B B B       A A A A A A A A A A A A B B B B
D D D E E E E E E E E E B B B B       D D D E E E E E E E E E B B B B
D D D E E E E E E E E E B B B B       D D D E E E E E E E E E B B B B
D D D E E E E E E E E E B B B B       D D D E E E E E E E E E B B B B
D D D E E E E E E E E E B B B B       D D D E E E E E E E E E B B B B
D D D C C C C C C C C C C C C C       D D D E E E E E E E E E B B B B
D D D C C C C C C C C C C C C C       D D D C C C C C C C C C C C C C
```

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