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The Graceful Tree Problem

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Draw some circles, the vertices, connected by lines, the edges. Make sure there are no loops in the structure. These are your “trees” or “insectoids” for the puzzle. Next: Fill the circles in with consecutive odd numbers, then write their differences on the connecting lines. Numbers should not repeat. This is the Graceful Tree Conjecture or the Ringel-Kotzig conjecture.

In this Numberphile video, Canadian board game and puzzle designer Gordon Hamilton of Math Pickle demonstrates how the anatomies of these different species help determine if they’re solvable or not.

He also discusses the most challenging versions of the Graceful Tree Conjecture. Hamilton uses this math puzzle in elementary school classes to help strengthen students’ subtraction and problem-solving skills.

graceful tree conjecture
graceful tree conjecture
graceful tree conjecture
graceful tree conjecture
We also recommend these math activity books: Bedtime Math and One Minute Mysteries: 65 Short Mysteries You Solve with Math!

Next: Anastasia Chavez explains the 21-card trick, Universal Calendar Puzzle: Figure out the day of the week for any date ever, how to multiply numbers by drawing lines, and can you solve the penniless pilgrim riddle?

Plus: How high can you count on your fingers?

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