The figure on the right shows the top left-hand corner of the complete diagram. Note the symmetry which leads to the three measurements of $\frac{1}{2}$. Thus the diagonal of the square can be divided into three portions of lengths:

$\frac{1}{2}$, $x$ and $ \frac{1}{2}$ respectively.

The length of the diagonal $= \sqrt{10^2 + 10^2} = \sqrt{200} = 10 \sqrt{2}$.

So $x$ = $10\sqrt{2} - 1$.

*This problem is taken from the UKMT Mathematical Challenges.*