The King requests Pythagoras to his palace to debate an vital matter.
After the same old formal greetings the King asks:
– I’ve been instructed that you’ve got a fabulous system about including squares collectively.
Pythagoras:
– Highness, what I found is that in a proper triangle with sides A, B and C, in case you add the sq. of A to the sq. of B, you get the sq. of C.
– So you’ll be able to add two squares and make a sq.?
– Effectively… Excellence, it is not precisely that, however there are in truth geometrical constructs to decompose two squares and type a single one.
– Good. Right here is the factor. I’ve a big quilt and a smaller quilt, each sq.. I would like you to mix them into a big sq. quilt.
– Actually, Best. I can lower this sq. diagonally, lower a triangle there, flip it round and transfer it up, …
– No, no, no! You possibly can’t do triangles. You possibly can’t lower diagonally. Cannot you see the quilts are made from small sq. photos of our goddess? These photos should stay intact and property oriented.
– I see. Your Magnificence is most fortunate, as a result of this quilt is 12×12 squares, that smaller quilt is 5×5 squares. That is 144 squares plus 25 that makes 169 squares, and that’s precisely what you want for a quilt of 13×13 squares.
I can cut up the small quilt into 25 squares and stich them on two sides of the big quilt, making a bigger one.
– That’s too many items. The stitching might be completed as per your directions however by skilled tailors. And also you see, their value is predicated on the variety of items they put collectively, whatever the size of the seam. With all these items they may get a fortune for little work. That can spoil them.
Inform me, what’s the smallest variety of items you could lower these quilts into, in order that they are often stiched again into a bigger sq. quilt, with all squares intact and property oriented?
– … I believe I might want to assume.
TL;DR
You have got a sq. of 12×12 unit squares and one other of 5×5 unit squares. Reducing alongside the gray strains, you need to cut up these into N items and, with out rotating or flipping anybody, simply transferring them, type a 13×13 sq..
What’s the smallest attainable N, i.e. the smallest variety of items?
Present how it may be completed.
