# Seven 2:1 rectangles covering a square

Can you fully cover a square with 7 rectangles such that:

1. Every rectangle has 2:1 ratio, ie., length double its width.
2. No part of any rectangle is outside the square.
3. No two rectangles overlap.

Note that rectangles can have different size. This puzzle is from a Numberphile video (see link in comments) and is possibly well known, but I haven't seen it here.

Is it possible?

Yes

Why?

6x6 is the smallest possible area that can be divided into 7 possibly distinct X by 2X rectangles (36 = 18 + 8 + 5*2). And the following layout works:

 AAAAAA
AAAAAA
AAAAAA
BBBBCC
BBBBDD
EEFFGG


Perhaps the intent is that the rectangles must all be different sizes, which would make it less trivial to solve.

• Correct and well done! Ok can you do it when all rectangles are different size? Jan 7, 2022 at 22:03
• @DmitryKamenetsky: The video you link to explicitly allows matching sizes. It starts with dividing a square into two $1 \times 2$ rectangles of the same size. Jan 8, 2022 at 5:46 