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Can you divide a 4x7 rectangle into 7 rectangles all of different area? Can you find multiple solutions?

Good luck!

P.S. @Deusovi wanted me to make puzzles that have an "aha moment", so here is my attempt at that.

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  • $\begingroup$ A better/ more optimal answer has been given. Please consider transferring the check to that answer, since it fits the solution better. Nevertheless, you have the final decision: this is only a reminder. Happy puzzling! $\endgroup$ – Omega Krypton Oct 13 at 23:34
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For completeness, I found all the solutions with integer-sided rectangles. As already noted, with 7 areas totalling 28 you are constrained to use areas 1 through 7, and only areas 4 and 6 have two possible sizes. So running that through my solver I get 272 solutions (excluding rotations/reflections).

The solutions:

enter image description here

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  • $\begingroup$ Thx Omega Krypton, I forgot how to do spoilers $\endgroup$ – theonetruepath Oct 13 at 23:35
  • $\begingroup$ Great work! Thank you for such a nice answer. $\endgroup$ – Dmitry Kamenetsky Oct 13 at 23:48
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This is,

possible.

Method:

First row: 1x7 rectangle,
Second row: 1x6 and 1x1 rectangles,
Third row: 1X5 and 1x2 rectangles,
Fourth row: 1x4 and 1x3 rectangles,
As for the second question, permutations of rows would give other solutions.

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Let's first work out the sizes of the rectangles.

The only way to partition $28$ into 7 different positive integer values is $1+2+3+4+5+6+7$.
The $4$ rectangle can be $1\times4$ or $2\times2$, and the $6$ can be $1\times6$ or $2\times3$. The others all have width $1$.

It turns out every combination of these shapes is possible.

 7 7 7 7 7 7 7
 6 6 6 6 6 6 1
 5 5 5 5 5 2 2
 4 4 4 4 3 3 3
 
 7 7 7 7 7 7 7
 6 6 5 5 5 5 5
 6 6 4 4 3 3 3
 6 6 4 4 2 2 1
 
 7 7 7 7 7 7 7
 6 6 5 5 5 5 5
 6 6 4 4 4 4 1
 6 6 3 3 3 2 2
 
 7 7 7 7 7 7 7
 6 6 6 6 6 6 1
 4 4 3 3 3 2 2
 4 4 5 5 5 5 5

Note that any sub-rectangle in these partitions can be rotated or flipped to get an alternative arrangement.

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