# Is there only one solution to the "Ten Penny Puzzle" or more?

Not knowing the dimensions of the square in the Ten Penny Puzzle I ask if there is more than one unique solution that isn't a reflection/mirror of the one shown in the video, if you know the dimensions of the square of course. Perhaps more interestingly though, I want to ask what if we have ten unit circles, then vary the bounding square around it from having a side length, $$s$$, of $$3$$ units ($$s\in\mathbb{R}$$) to see the minimum side length it takes greater than $$3$$ units to generate a solution (possibly not singularly unique). Also, how many unique solutions exist in the range $$s\in(3,4)$$?

The puzzle involves fitting 10 pennies into the square shown without overlapping:

• Could you include a full description of the puzzle in your question so that we don't have to watch the video? Apr 7, 2018 at 7:40

## 1 Answer

With regards to the uniqueness of the solution:

The solution to the puzzle shown in the video is the most optimal packing of 10 circles in a square, so assuming that the dimensions of the puzzle are tight enough, then the solution is unique.
For a proof of optimality of that solution, you can read the following paper:
The Optimal Packing of Ten Equal Circles in a Square
by Claas de Groot, Ronald Peikert, and Diethelm Würtz

For other solutions:

You can see (near) optimal packings with other numbers of circles on the Circle packing in a square Wikipedia page. By leaving out some of the circles in the solutions with 11-15 circles you get lots of possibilities for packing 10 circles in a square of size $s<4d$ where $d$ is the circle diameter.