Puzzle: Perfect rectangles, tiled with distinct squares, squaring the rectangle

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Solutions to Perfect rectangles nrs 1 to 10

All solutions are found with backtracking. They are ordered by the length of the long side. Only original (simple) solutions have got a number.

====================== 33x32 ======================

01. 32 x 33 rectangle, with only 9 squares
This is by far the smallest possible rectangle that can be filled perfectly with squares. This solution is known for many years. It seems proven that any solution exists of at least 9 squares... Who can prove this?
How many primary solutions exist of this minimum of 9 squares?

====================== 57x55 ======================

02. 57 x 55 rectangle, with 10 squares
Nearly twice as long as the first one, hard to believe we couldn't find any solution between.
The 1x1 piece is not used in this solution

====================== 65x47 ======================

03. 65 x 47 rectangle, with 10 squares too, how many 10-ers exist?
After quite a while we found this oblong solution. Are there any which are more oblong?
The 1x1 piece is not used in this solution

============== 65x32, 65x33 en 65x64 ==============

65 x 32 en 65 x 33 can be solved too, not printed because they are but not primary, they are both the the first step in the fibonacci-spirals out of the 33 x 32 rectangle.

The 66 x 64 can be solved too, but this is the double version of 32 x 33.

====================== 69x61 ======================

04. 69 x 61 rectangle, another one with the minimum of 9 squares!
After many non-original solutions we continue with e real one.
The 1x1 piece is not used in this solution

====================== 75x73 ======================

05. 75 x 73 rectangle, with 16 squares
This solution uses a large number of squaers.

====================== 79x65 ======================

79 x 65 rectangle (not primary)
Combination of other solutions and a square. However it's not a real original solution, I found it too curious to omit.
It's the first fibonacci-step of 32x33, coupled to the 65x47 solution.

====================== 79x74 ======================

06. 79 x 74 rectangle, with 16 squares
One more with length 79! The largest pieces is relatively big, compared to other solutions.

====================== 81x80 ======================

07. 81 x 80 rectangle, with 12 squares
This solution is nearly square, there are many N by N-1 solutions like this one.

====================== 82x75 ======================

08. 82 x 75 rectangle, with 17 squares

====================== 83x77 ======================

09. 83 x 77 rectangle, with 14 squares

====================== 84x60 ======================

10. 84 x 60 rectangle, with 17 squares
One of the most oblong upto now, hard to solve with your bare hands.

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Solutions 11 upto 20

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