# Cut this shape into 3 pieces and fit them together to form a square

A shape is drawn on a sheet of squared paper as shown in the picture below.

The shape is then cut from the sheet and given to you. You are asked to first make a straight cut across the shape and then make a straight cut across any of the two resulting pieces so that the three pieces you end up having fit together to form a perfect square. You are not allowed to make folds, overlaps, or any other cheats, and there's no trick in the wording. Just cut and see whether you can make a square.

Can you crack this puzzle?

• If we do this to what degree of accuracy is meant by perfect square? Like what is close but not perfect square and what is correct? It seem like sense we can not see it beyond say millimeter it like guessing and then the probability of guessing one number out of any non empty set of necessarily real numbers is zero and thus this puzzle is impossible. Apr 6, 2020 at 10:51
• @marshalcraft You're really overthinking this puzzle. Generally, if something is overlaid onto a grid, you can assume that grid to be accurate. Apr 7, 2020 at 12:37

First cut in red, second in blue. The green shape is the final 5x5 square.

We know the final square must be 5x5 as there are 25 total squares.

A slightly different graphic:

Becomes

• Yes, it works, although my solution is very different :) Apr 6, 2020 at 0:53
• @Mitsuko I will keep looking for other answers then :) Apr 6, 2020 at 0:59
• @Mitsuko I’m working on a theory that the intended answer involved diagonals, would I be right in thinking that? Apr 6, 2020 at 1:14
• Yes :) When I invented this puzzle, I overlooked that it has a more straightforward solution :) Anyway, you was first to post a correct answer, so I accepted your answer :) Apr 6, 2020 at 5:20
• Interestingly, this solution uses a particular property of the side lengths $(a,b)=(3,4)$ of the smaller squares: for the pieces to fit together right, we need $a+b/2 = \sqrt{a^2+b^2}$. So for general side lengths, this method wouldn't work and the less rectilinear intended solution might be required. Of course, any rectilinear dissection requires that the total area be a perfect square.
– xnor
Apr 6, 2020 at 11:34

Here is a solution that works in the general case of two squares of any size placed next to each other.

• Yes, this is the solution I had in mind :) Apr 6, 2020 at 5:30
• Technically this fails for some, say squares 2x2 and a 3x3. But there is reason why that is kind of trick. But if this method failed it would leave puzzler to have to explore additional things until trick answer realized. Apr 6, 2020 at 9:42
• @marshalcraft I don't see how it would fail with squares of size 3x3 and 2x2. Apr 6, 2020 at 9:52
• ... 3x3 square is 9 blocks a 2x2 is 4 blocks for a total of 9+4=13 blocks. 13 is a prime number a thus there is no way to make a square out of the combined blocks. It works when there are solution to the puzzle and the case that was asked. But I just said for trick cases. Apr 6, 2020 at 10:06
• @marshalcraft Are you referring to Beastly Gerbil's answer rather than mine? My answer with diagonal cuts works perfectly well with 13 blocks. The cuts would be of length $\sqrt{13}$. Apr 6, 2020 at 10:10