TITLE: Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes
QUESTION [6 upvotes]: It is know that a square can be dissected into other square such that no two of the squares have the same size.
This is the simplest dissection of that kind:

Is it also possible to dissect an equilateral triangle into equilateral triangles such that no two of them have the same size?

REPLY [2 votes]: Although as Ross Millikan points out, such a dissection is impossible, it is of interest to note that, provided we count upwardly oriented triangles as different from downwardly oriented, there is a dissection into 15 triangles. That is, none of the upwardly oriented triangles are same sized, nor any of the downwardly oriented.
http://arxiv.org/abs/0910.5199
[Note that site allows a free PDF download.]