1. Divide a square into three similar regions, ALL THREE of which are congruent.
2. Divide a square into three similar regions, EXACTLY TWO of which are congruent.
3. Divide a square into three similar regions, NO TWO of which are congruent.
4. Divide an equilateral triangle into three similar regions, ALL THREE of which are congruent.
5. Divide an equilateral triangle into three similar regions, EXACTLY TWO of which are congruent.
6. Divide an equilateral triangle into three similar regions, NO TWO of which are congruent.

If necessary, you may use one or more inversely similary regions where all corresponding angles are equal and described in the opposite rotational sense, i.e. "reflected".

I've thought about this kind of problem (with squares anyway) in the past.  This idea can be extended to more similar pieces and combinations of congruences.

Each type of solution can be noted as in a partition

http://mathworld.wolfram.com/PartitionFunctionP.html

In this probem part 1 is 3, part 2 is 2+1, and part 3 is 1+1+1.  These are the partitions of 3.

The partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.
These are pretty easy with a square, but I can't find 3+1 or 2+1+1 with a triangle. 

I'm convinced that any partition of squares into similar rectanges is possible.  Based on my 2+1 triangle solution, the triangles solutions tend to get really complicated if they are possible at all.

 

Posted by Jer on 2006-02-16 13:33:42