On Wed, 6 Apr 2005, Philippe 92 wrote: > Canon wrote : >> A good place to start for online information is: >> http://mathworld.wolfram.com/Dissection.html > > Sure, but this problem is slightly different. > We don't want to dissect a polygon into another one, > but to "bissect" a shape in two identical (congruent) shapes, > each one being only one piece. > Seems there is much less litterature about this. > > Releasing the 1 piece per polygon constraint results also into > interesting puzzles : > > Dissect a square so as to build 3 equal squares, 6 pieces. > Dissect a square into two equal squares is trivial.
(MathWorld provides an example of dissecting a square into two /unequally/ sided squares, under "Pythagorean Square Puzzle.")
I'm sure a lot of rec.puzzlers haven't (recently?) seen the "dissection" of a square into a number of squares of unequal sizes, listed on MathWorld under "Perfect Square Dissection" (a.k.a. "squaring the square"). For example, a square with side length 112 can be dissected into 21 pieces, each of which is a square, and no two of which are of the same side length. Larger perfect square dissections also exist.
Puzzle: Is it possible to produce a "perfect triangle dissection," that is, to dissect an equilateral triangle into K>1 similar equilateral triangles, no two of which are of the same side length? If so, provide an example; if not, why not? (I just saw this problem, and solution, today in a course on graph theory. Hints available on request.)
Open-ended (computer?) puzzle: It is possible to produce a perfect NxM-rectangle dissection (into similar but non-congruent rectangles) simply by stretching a perfect square dissection in one of its dimensions. But can you find the smallest perfect NxM-rectangle dissection /not/ obtainable by this method? Let's call the dissection "smallest" that minimizes M+N+K, where M>N>0 are integers. What other interesting perfect dissections can you find? Is there a perfect dissection of the "P" pentomino?