Two unit squares overlap with an area 1/16. The shape of this overlap may be a triangle or a quadrilateral.

Find the minimum distance between their centers.

Although this is under the *Calculus* heading, it was unclear to me

how to use calculus to solve it. The overlap is governed by three

parameters of the overlapping square: its rotation around its

center, its delta_x, and delta_y wrt to the fixed square.

I solved this problem using a grid search. Delta-y was ranged between 0 and (1+sqrt(2))/2 and theta from 0 to 45 deg. For each pair, I found

the delta_x that gave the required overlap and using these solutions, then

found the minimum separation out of all the cases produced.

A rough grid is shown

*here*.

The trough for the minimum separation is broad and shallow and centered

on theta = 9 +/- 2 degrees, delta x=0.921 and delta y=0.16

with x and y uncertainty of about 0.02.

The rough grid has some uncertainty in its resulting parameters. But more

refined grids gave consistently a minimum separation of d=0.934.

The (0.16, 0.921, 9 deg, d=0.935) configuration looks like the

*this.* The solution overlap was a three sided shape, but many overlaps that came close had four sides such as **this**. (0.10, 0.935, 12 deg, d= 0.943)

*Edited on ***May 31, 2023, 10:05 pm**