Vietnamese professor cracks 55-year-old math puzzle, inspired by juggling
A Vietnamese mathematician and his German colleague have helped resolve a 55-year-old mathematical conjecture, using randomness and techniques inspired by juggling to tackle the problem.
In February, Professor Pham Tuan Huy and mathematician Lisa Sauermann published a 27-page proof addressing a 55-year-old math puzzle posed by Ronald Graham in 1971. The conjecture questions whether a set of distinct nonzero numbers can always be rearranged so that all partial sums, or the sum of the first number, the first two, the first three, etc., are different.
In juggling terms, this means rearranging throws so no two balls land on the same beat. If all numbers are positive, the answer is trivially yes, but the problem becomes harder with repeating numbers. The mathematicians became interested in the problem after a conference in Germany and developed a complex method using anticoncentration, a technique from Fourier analysis, to show that random sums are spread out, making undesirable configurations unlikely.
By estimating the probability of each undesirable outcome, they proved at least one valid arrangement must exist. Their work, combined with earlier results, completes the conjecture for all sufficiently large primes.
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