VWO 1993 Problem 2

A jeweler covers the diagonal of a unit square with small golden squares in the following way:

• the sides of all squares are parallel to the sides of the unit square
• for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex)
• each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$ )
• all midpoints are on the diagonal

(a) What is the side length of the middle square? (b) What is the total gold-plated area?