You have a square piece of paper and you do not know its length or width.

What is the minimum number of folds required in order for the paper to only show 1/16 of its original size, 2/16 of it's original size, 3/16 of its original size........15/16 of its original size.

Note: You do not know where halfway marks or quarter marks lie on this piece of paper therefore you must use folds to mark these areas. You are allowed to use folds as guides to make other folds.

I would like to propose a solution, using just squares & rectangles (and the occasional triangle). The solution should be flat, I considered 3d topology, but it got too hard (and I was running out of paper :-) )

1) To get from x/16 from 2x/16, it takes one fold.
2) Upon breaking up x into binary digits, and writing it everything out. I'm pretty sure that one can see a simple pattern. I have rearranged the digits to better reflect the pattern.


...............(Squares)..(Triangles)
08/16-1000-1 fold
04/16-0100-2 folds
12/16-1100-2 folds
02/16-0010-3 folds
06/16-0110-3 folds....2 folds
10/16-1010-3 folds
14/16-1110-3 folds
01/16-0001-4 folds
03/16-0011-4 folds....3 folds
05/16-0101-4 folds
07/16-0111-4 folds
09/16-1001-4 folds
11/16-1011-4 folds
13/16-1101-4 folds
15/16-1111-4 folds

I believe you can do better with triangles for some of them, maybe someone out there can help.

Posted by delvin on 2003-02-12 16:50:45