by zkfcfbzr (original post)
Let \(l\) be the length of the shaded squares.
It follows that the diagonal of the unit square is equal to
$$ \begin{align*} \sqrt{2}&=2\sqrt{l^2+l^2}+l\\ &=l(2\sqrt{2}+1) \end{align*} $$
Solving for \(l\)
$$ \begin{align*} l&=\frac{\sqrt{2}}{2\sqrt{2}+1}\\ &=\frac{\sqrt{2}(2\sqrt{2}-1)}{(2\sqrt{2}+1)(2\sqrt{2}-1)}\\ &=\frac{4-\sqrt{2}}{7} \end{align*} $$
\(\therefore a+b=11\)