The King requests Pythagoras to his palace to discuss an important matter.
After the usual formal greetings the King asks:
- I have been told that you have a marvelous formula about adding squares together.
- Highness, what I discovered is that in a right triangle with sides A, B and C, if you add the square of A to the square of B, you get the square of C.
- So you can add two squares and make a square?
- Well... Excellence, it's not exactly that, but there are in fact geometrical constructs to decompose two squares and form a single one.
- Good. Here is the thing. I have a large quilt and a smaller quilt, both square. I need you to combine them into a large square quilt.
- Certainly, Greatest. I can cut this square diagonally, cut a triangle there, turn it around and move it up, ...
- No, no, no! You can't do triangles. You can't cut diagonally. Can't you see the quilts are made of small square pictures of our goddess? These pictures must remain intact and property oriented.
- I see. Your Magnificence is most lucky, because this quilt is 12x12 squares, that smaller quilt is 5x5 squares. This is 144 squares plus 25 that makes 169 squares, and that is exactly what you need for a quilt of 13x13 squares.
I can split the small quilt into 25 squares and stich them on two sides of the large quilt, making a larger one.
- That is too many pieces. The sewing will be done as per your instructions but by expert tailors. And you see, their price is based on the number of pieces they put together, regardless of the length of the seam. With all these pieces they will get a fortune for little work. That will spoil them. Tell me, what is the smallest number of pieces that you can cut these quilts into, so that they can be stiched back into a larger square quilt, with all squares intact and property oriented?
- ... I think I will need to think.
You have a square of 12x12 unit squares and another of 5x5 unit squares. Cutting along the grey lines, you want to split these into N pieces and, without rotating or flipping any one, just moving them, form a 13x13 square.
What is the smallest possible N, i.e. the smallest number of pieces?
Show how it can be done.
It is less than 6.
I mention 6 because I found a number of different solutions with that count.