A unit square has a fixed position. A second unit square overlaps the first. The overlap is a rectangle with area 1/2.
Find the area enclosed by the locus of the center of the second square.
Except for one final calculation, this problem is a Paper and Pencil problem for a Calculus student.
I will focus on one quadrant of the locus, then multiply by 4.
Place the fixed square centered at the origin. Let the other square be centered at (x,y). Then a pair of diagonally opposing corners of the overlap are at (1/2, 1/2) and (x-1/2,y-1/2).
The area can then be written as (x-1/2 - 1/2) * (y-1/2 - 1/2) = (x-1)*(y-1). This equals 1/2 by the problem statement.
So equate and solve for y to get y = 1/(2*(x-1)) + 1. Integrating this over 0 to 1/2 will give us the area of a quadrant of the locus area.
The antiderivative is ln|x-1|/2 + x. Evaluating this at 0 to 1/2 gets a value of (1/2)-(ln 2)/2.
Multiply this by 4 to get a final answer of 2-2*(ln 2) ~= 0.6137.
Solution
