# 4x7 rectangle divided into 7 different rectangles

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.

• 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! – Omega Krypton Oct 13 '19 at 23:34

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:

• Thx Omega Krypton, I forgot how to do spoilers – theonetruepath Oct 13 '19 at 23:35
• Great work! Thank you for such a nice answer. – Dmitry Kamenetsky Oct 13 '19 at 23:48

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.

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.

• Very well done! – Dmitry Kamenetsky Oct 13 '19 at 6:00