All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Splitting shapes (Posted on 2006-02-16) Difficulty: 3 of 5
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".

See The Solution Submitted by Bob Smith    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts going further | Comment 3 of 9 |

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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information