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Neurobics

Folding and Cutting

Russell Dear

Take a piece of paper (the size is not important but it shouldn't be too small), fold it in half and cut off the corners of the resulting rectangle.

Fold the paper in half again with a crease at right-angles to the first and remove the uncut corners.

Repeat the process, each time folding the paper perpendicular to the previous fold, and cutting off the corners of the resulting shape.

You'll find you won't be able to fold more than six or seven times. Now, if after each time the corners were cut off, the paper was unfolded, you would find some holes in it. For example:

No. folds|1|2|3|4
No. holes|0|1|3|9

The question is, can you predict the number of holes when the paper is folded five times? How about six times? Seven times? Ten times?

Russell Dear is a Mathematician living in Invercargill