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.


2 Answers 2


Is it possible?



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:


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

  • $\begingroup$ Correct and well done! Ok can you do it when all rectangles are different size? $\endgroup$ Jan 7, 2022 at 22:03
  • $\begingroup$ Here is the cool Numberphile video about this puzzle: youtube.com/watch?v=VZ25tZ9z6uI $\endgroup$ Jan 8, 2022 at 0:51
  • $\begingroup$ @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. $\endgroup$ Jan 8, 2022 at 5:46
  • 1
    $\begingroup$ @Ross yes the original question allowed duplicate sizes. Now we are making it harder for fun :) $\endgroup$ Jan 8, 2022 at 6:14

By visual inspection of the diagram below it can be seen that a 3x3 square is divided into 7 rectangles whose sides are in 2:1 ratio.

rectangles within square


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.