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The Checkered Origami Triangle

Checkered Patterns

by Serhiy Grabarchuk

 

An Origami triangle.

 

How to get a 2-unit-side checkered triangle.

 

Three 2-unit-side checkered triangles.

 

Fifty-nine 3-unit-side checkered triangles.
 
 
This is a triangular version of the Origami Checkerboard puzzle. It is also a rare triangular Origami work since it uses a sheet of paper in the form of an equilateral triangle colored just on one of its sides.

A small sample at left shows how to get a 2-unit-side triangle with one colored cell.

In this page you can see three, 2-unit-side patterns, and fifty-nine 3-unit-side patterns. Your challenge is to achieve every of the proposed patterns in the minimum of single "book" folds. The final shape every time is a flat equilateral triangle; no matter what pattern will be at its back. Numbers next to the patterns indicate the minimal known numbers of folds necessary to perform them. Write us if you can improve any of these results.

Pattern 6-U, solution to it, and solutions to patterns 3-c, 4-I, 5-W, 5-X, 5-Y, 5-Z, 5-AB, 5-AC, 6-L, 6-M, 6-N, 6-O, 6-Q, 6-R, 6-S, and 6-T were found by Keiichiro Ishino. A solution to pattern 5-AA was found independently by Setsuo Sasaki and Keiichiro Ishino. Solutions to patterns 3-b and 3-J were found by Hirokazu Iwasawa and Naoaki Takashima, respectively.

***
Much more difficult challenges can be finding all the 4-unit-side checkered triangles and, then, finding their minimal solutions. Recently, Keiichiro Ishino calculated that there are 6,605 substantially different 4-unit-side checkered triangles. Part of them have their solutions quite similar to the respective 3-unit-side checkered triangles described here, but it is obvious that most of them are rather difficult to be optimized.
 
 
Last Updated: October 14, 2008
Posted: September 4, 2005
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