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You may wish to look at Folding squares before trying this problem.
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Folding Fractions
Age 14 to 16
Challenge Level 
You may wish to look at Folding squares before trying this problem.
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You can divide the long diagonal of a square into different fractions by folding.
This problem is about the fractions of the long diagonal of a square which you can construct in this way.
To start with, we shall only consider points on the side of the square which can easily be found by folding. That is, $\frac{1}{2}$s or $\frac{1}{4}$s or $\frac{1}{8}$s and so on.
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Investigate the fractions of the long diagonal of a square that can be created in the way described above. Here are some examples to think about:
... ... Can you extend the findings and make generalisations?
Can you justify your generalisations?
What about starting with fractions of the side of the square that are not so easily found by folding?
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