# Dissection: $7^2 + 1^2 = (5\sqrt{2})^2$ and related problems

What is the minimum number of pieces needed to dissect a $$7 \times 7$$ square, plus a $$1 \times 1$$ square tile, to form a $$5\sqrt{2} \times 5\sqrt{2}$$ square? The $$7 \times 7$$ may only be cut by line segments either parallel or at 45 degrees to its sides, and the $$1 \times 1$$ may not be cut.

For this minimum number, can the pieces be rearranged without reflection? If so, without rotation?

Although it seems easiest to require the cuts to intersect integer or rational points, which would ensure that the final square be tilted by 45 degrees, I would also encourage thinking about irrational placements, which could create e.g. a diagonal segment with length $$1$$.

The best solution I know of consists of six pieces (including the $$1 \times 1$$) with reflection:

There are similar problems based on other rational approximations to $$\sqrt{2}$$, e.g. $$41^2 + 1^2 = (29\sqrt{2})^2$$. How does the number of pieces for these grow? Below is one of a family of translation-only solutions that works for both underestimates like $$\frac{7}{5}$$ and overestimates like $$\frac{3}{2}$$, which can be seen in the center. (The grid lines are spaced half a unit apart.)

However, maybe the two subfamilies of problems should be considered separately, since depending on the subfamily, the $$1 \times 1$$ is added to either the orthogonal square or the diagonal one, e.g. $$(12\sqrt{2})^2 + 1^2 = 17^2$$.

This is my first post, so please let me know if it should be split into multiple questions or if there are any other issues.

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• Distant cousin: Pythagorean Pentagons Commented Aug 2 at 0:29