Now, change the task: Dissect a rectangle into three congruent pieces.
Can it be done in more than one way?

And finally, shift the problem using these modifications:

- Dissect a rectangle into n congruent pieces, for n>3
- Dissect other quadrilaterals (e.g. non-rectangular trapexoids, parallelograms, kites, general convex or concave quadrilaterals) into 2, 3, ... congruent pieces. Possible?
- Dissect a rectangular parallelpiped (i.e. box) into 2, 3, ... congruent pieces. Possible?
- What about dissecting other 3-dimensional and/or n-dimensional shapes into congruent pieces?

**Hint:** Hints have already been provided for the case of dissection into two congruent shapes...

**Solution Commentary:** Perhaps you arte frustrated trying the case of three congruent sections of a rectangle....but hopefully it was fun trying!

In 1994, Samuel Maltby proved: **If a rectangle is dissected into three congruent pieces, the pieces themselves must be rectangles.** [*Journal of Combinatorial Theory*, 66: 40-52].

His proof is lengthy and very technical, but uses only elementary mathematics involving generalizations akin to working with polyomino tiles.

As to the other suggested generalizations to 2-dimensional quadrilaterals and those of higher dimensions, I believe they are still unsolved problems.