Today’s puzzle is about transport links and soapy water.
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?
Here is a wrong answer. The O-loop uses is too much unnecessary asphalt.
The U-route is also needlessly lavish.
The X-road is shortest way to get from any town to the one in the opposite corner, but is also not the optimal answer for overall length of road.
Can you work out what the minimal road network looks like?
I’m not expecting you to work it out mathematically – that requires very sophisticated techniques. But you might be able to make an intuitive guess.
Alternatively, you can solve this problem with 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 place it in a bowl of soapy water, bubbles will form around the dowels displaying the answer.
Nature finds the minimal structure instantly. The plastic model is like an analogue computer. In fact, here’s a hint: the solution resembles a simple geometrical shape that appears in the real world in a very familiar setting.
I’ll be back at 5pm UK with the answer and a video of the soap bubble solution.
PLEASE NO SPOILERS Please discuss your favourite bubbles.
I have been meaning to share this puzzle for ages, and was recently reminded of it when I went to MathsWorld London, a maths discovery centre that has just opened in Southwark, a few minutes walk from the Tate Modern.
The venue has dozens of interactive exhibits and one of the ones my kids loved the best was the gigantic soap bubble machine. You stand in it and pull a bubble over your heads.
Bubble double: children at MathsWorld London
Other exhibits include an elliptical pool table, a Morse code machine, a 5ft build-your-own arch and many hands-on puzzles.
MathsWorld London is the capital’s first attraction uniquely focussed on maths and was the culmination of years of planning and fund-raising. Each exhibit reflects some mathematical idea through play, with explainers at hand if you want to know more. It’s a brilliantly inclusive and joyous space, and a welcome addition to London’s cultural landscape.
If you don’t live in London, there’s also a giant bubble machine in MathsCity Leeds, a sister project to MathsWorld London which has been running since 2021.
MathsWorld London is at 6 Burrell St, London, SE1 0UN.
MathsCity Leeds is at Zurich House, 4 Canal Wharf, Leeds, LS11 5PS.
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.
