One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
The Perfect Cuboid — Can All Seven Lengths Be Integers?
Three edges, three face diagonals, one space diagonal. Make all seven integers. No such box has been found in 132 years since the published 1894 challenge.
Learn the rule, then go farther with calculation or a small program.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesSee the puzzle visually
An Euler brick gets the six outer lengths right. No box is known whose seventh length through the interior is also an integer.
Problem statement
Do positive integers exist such that all three face diagonals and the space diagonal are integers as well?
The story of this puzzle
The Pythagorean theorem tells us how to make one rectangular face integral. The challenge is to succeed on all three faces simultaneously and then make the long diagonal through the box integral too. In 1719 Paul Halcke recorded the smallest Euler brick, with edges and face diagonals . Six lengths worked; the space diagonal did not.
In 1894 Artemas Martin published the challenge of filling that final slot. Since then people have built formulas and searched enormous ranges by computer. Modern searches imply that any solution must have an odd edge beyond and a smallest edge beyond . Yet absence from a huge range is not proof of absence forever.
Even near misses are worth collecting. Compare boxes where two face diagonals work but the third narrowly fails, or Euler bricks where all six outer lengths work and only the space diagonal fails. The search for one perfect box and the proof that none can exist begin from the same experiments.
Try it yourself
Check the box. Verify by Pythagoras that has face diagonals 125, 244, and 267. Then compute the space-diagonal square and locate it between consecutive squares. How narrowly does the seventh integer fail?
Beyond MathVoyage
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