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Brought to you by: Tom "Licensed to Fold" Hull
Take a square piece of paper that is white on one side and colored on the other, and fold it so that the colored side makes the pattern shown to the right.
This can be done in only five folds. (Here we are counting only the folds used, not folds we might need to get exact landmarks.) Really! If you don't believe me, click here to see how.
According to my sources, this type of puzzle was originally created by Serhiy Grabarchuk of the Ukraine. Some Japanese puzzle masters, in particular Kozy Kitajima and Hiroshi Yamamoto then expanded the idea and found numerous solutions. Below are some of their puzzles, together with the minimum number of folds required.
Can you figure out how Kitajima-san and Yamamoto-san make these designs using so few folds? Can you find ways to do them in even fewer folds? Email me if you do!
Area Puzzle 1:
In his book Mathematical Brain Benders (Dover, NY, 1982) Stephen Barr presents the following puzzle:
Starting with a square sheet of paper, fold it to produce a square having three-fourths its area. Only five folds are allowed.
In 1994 I posted this puzzle on the internet origami mailing list, claiming that it could be done in only four folds. Can you do it?
Note: In this puzzle, every step counts as a "fold". For example, to find the center of the paper requires two folds (two book folds, say). If you're having trouble getting started, here's a hint.
Area Puzzle 2:
Fold a square to produce an octagon that is 1/6 the area of the original square.
I can do it in six folds. Can you do better? Email me if you can! (Sorry - no hints for this one!)
Area Puzzle 3:
If you figured out (2), try this: fold a square to produce an octagon that is 1/3 the area of the original square.
This is a bit more tricky - even when you have the right answer it's quite hard to prove that the area is 1/3. This one takes me eight folds.
Can you think of other nice area puzzles? How about folding a square that is 1/5 the area, or 2/5 the area?
I am indebted to Edward Kitchen's article "Dörrie tiles and related miniatures" (Mathematics Magazine, Vol. 67, No. 2, April 1994, pp. 128-130) for providing the inspiration for puzzles (2) and (3).
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