Graceful Trees

Can you arrange five consecutive odd numbers in a line, so that the differences between adjacent pairs of numbers are all different? What if the numbers are connected in a way other than in a line?

The Graceful Tree Conjecture in graph theory says that we will always be able to arrange n evenly-spaced numbers in a tree (with n nodes) so that differences between adjacent pairs are all different. Explore this with your students!

Curriculum Topics Covered

Graceful Creatures: Basic number facts, problem solving, thinking systematically, rotations and reflections, graph theory. Optionally: simple algebraic proof.

Difference Trees: Basic number facts,  problem solving, thinking systematically.

More Information

See Teacher Notes and Solutions for more information, ideas for class discussion and solutions to the problems. 

These activities were inspired by the Numberphile video ‘The Graceful Tree Problem’, featured below.