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Topic: Where can I find...
Replies: 2   Last Post: Apr 6, 2005 10:15 PM

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Arthur J. O'Dwyer

Posts: 91
Registered: 12/13/04
Re: Where can I find...
Posted: Apr 6, 2005 2:54 PM
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[xposted to sci.math]

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?

-Arthur



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