Can you solve it? Here's (not) looking at Euclid

The Guardian

17-03-2025

The ancient Greek geometer Euclid presented a list of five axioms he held to be self-evidently true. They are (or are equivalent to):
You can draw a line between any two points.
You can extend lines indefinitely.
You can draw a circle at any point with any radius.
All right angles are equal.
All triangles have internal angles that add up to 180 degrees.
Euclidean geometry is what we learn at school, and only applies to flat surfaces. The internal angles of a triangle on a curved surface do not add up to 180 degrees – the topic of today's puzzles.1. Right, Right, Right.
Assume the Earth is a perfect sphere. Imagine drawing a straight line from the North Pole to a point on the Equator. Can you draw two more identical lines to make a triangle where all the internal angles are right angles (i.e. they add up to 270 degrees overall)?2. Full circle
Next, let's go bigger, angle-wise. Can you find a way to cover the Earth with equilateral triangles that have internal angles of to 120 degrees (i.e. they add up to 360 degrees overall)? These triangles must all be the same size and there must be no overlaps or gaps between them.
(Hint: think about drawing triangles side by side.)
3. Tasty triangles
Now imagine a donut instead of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? And what will the sum of the six internal angles of these two triangles add up to?
(A donut is a 'torus', a cylinder that curves and joins itself in a loop, as in the image above.)
I'll be back at 5pm UK. PLEASE NO SPOILERS. Instead discuss your favourite axioms.
Today's puzzles were set by Adam Kucharski, who is a maths professor at the London School of Tropical Medicine and a popular science author.
In his brilliant new book Proof: The Uncertain Science of Certainty Adam tells the story of how nineteenth century thinkers began to challenge Euclid's self evident truths – and how this shaped the history of mathematics. It's a great read that covers many fields, including history, politics, statistics, computer science and epidemiology, which is Adam's area of professional expertise.
Proof by Adam Kucharski is out in the UK on Thursday and available at the Guardian Bookshop.
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.