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 · DiscoverMeet 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 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Magic Square of Squares — Can 3×3 Be Done?
Problem
Is there a magic square (rows, columns, diagonals all sum to the same value) whose nine entries are distinct positive perfect squares?
Why it matters
Even the origin needs a boundary: a 2025 peer-reviewed account traces a form of the question to Édouard Lucas in 1876, while Open Problem Garden starts with Martin LaBar in 1984. Martin Gardner popularized it with a $100 prize in 1996. A €1,000-and-champagne offer by Christian Boyer is also recorded in 2005, but surviving sources conflict on whether it targeted the original nine-distinct-positive-squares problem or an easier variant. A 7-of-8 near miss is not almost a proof.
Progress so far
Bremner (1996) and later computations produced 7-of-8 near misses and strong necessary conditions, but neither a square of nine distinct positive integer squares nor a nonexistence proof is known. A 2023 study further restricted possible prime-factor patterns. Search bounds depend on the formulation and should not be collapsed into one universal cutoff.
Further reading
💡 Explore together, one line at a time(0 contributions)
Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.
What did you notice?
You do not need a complete proof. A small observation can open the next path.
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