# An optimal puzzle (unit)

Given the image below, which of the options, A-E, should replace the question mark? Explain your reasoning.

Hint 1

I Initially planned to write "An optimized puzzle" but thought it would be too much of a giveaway. Let's see if someone can manage to figure out what this puzzle is about...

Each square contains a polygon whose vertices depict

points where circles would be centered to pack n unit circles in a square, for N = 1 through 7. These points have then been connected (roughly clockwise) using line segments. The number given below each square is the side length of the square which surrounds the unit circles.
For example, to place one unit circle in a square, the square must have side length of 2, and the circle is centered in the middle of the square.

Here is a visual for 2-6 (credit Wikipedia link above; case 1 is trivial and omitted)

The missing figure is for 7 unit circles, which looks like this:
, or B (~5.7).
My in to the puzzle was noticing that cases 1, 2, and 4 have vertices spread out in such a manner that seem to optimize the size of congruent circles around them. The linked article confirmed the rest.

• Wow there are some pretty pictures here. Commented Oct 9, 2023 at 17:24
• Great work! Nice to see it solved :) Commented Oct 9, 2023 at 17:25