Earlier today I set a puzzle which is extremely hard to answer if you are not a soap bubble.
The four towns
Four towns are situated at the corners of a square, as below:
What is the road system that connects all four towns using the smallest total length of road?
What perhaps feels like the right answer is a network where opposing towns are connected in straight lines.
If the square has side length 1km, the total length is about 2.83km
In fact, the minimal network is the one below, which shaves off about 4 per cent of the length of the X solution. The three angles at the intersection points are equal, 120°.
The right answer. Total length is about 2.73km.
Proving theoretically that this pattern is minimal requires advanced calculus. But we can find the answer more simply, using soap bubbles.
If you make a plastic model of the puzzle (a sandwich where the “bread” is two pieces of transparent flat material and the “filling” is four short dowels of equal length, positioned at the corners of a square) and dunk it in soapy water, bubbles will form around the dowels displaying the answer. Here’s a video of how it works presented by James Grime.
I love this problem because it is one of the most powerful illustrations of how Nature effortlessly solves optimization problems that might otherwise require a lot of hard thinking.
I gave a hint saying that the pattern resembles a geometric shape that appears in the real world in a very familiar setting. If the angles at the intersection points are all 120°, we get hexagons – which is Nature’s answer to how to most efficiently store honey. Sweet!
Photograph: florintt/Getty Images
For further reading about the science of soap bubbles, I recommend this classic 1976 article The Soap Film: An Analogue Computer from American Scientist.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
