An Eisenstein triple is related to 60 degree triangles and a special case of the cosine law. But we need not worry about that except to note that a specific example of an Eisenstein triple is $7^2 = 5^2 - 5\times8 + 8^2$ which we can rewrite as $7^2 + 5\times8 = 5^2 + 8^2$
Which means we can have a puzzle involving eight octominoes and five pentominoes, colored as per the given 9x10 rectangle with missing corner.
Arrange the five blue pentominoes into a 5x5 square
Arrange the five orange octominoes into a 5x8 rectangle
Arrange the five blue pentominoes plus the three pink octominoes into a 7x7 square
Arrange the five orange plus three pink octominoes into an 8x8 square.
All these have a single solution (ignoring rotations and reflections). You are allowed to flip pieces over. And to keep things simple the color on the other side is the same. Using a computer will just spoil it for you, these are at a 'hand tiling' difficulty level.
- Arrange all 13 pieces into a 9x10 with missing corner, like the diagram here, but in such a way that each color forms a single connected area. Touching at a corner is not touching in this instance. There are seven ways to do this (ignoring R&R as usual). If you feel that you haven't done enough, find all seven.