Solution to Problem 122

There were no correct solutions from Bradley University students. Solutions were submitted by alumnus Brian Laughlin ('81), and from J.D. Anderson, Jens Voss, Ron Welch, Christoph Fieberg, Lenny Gibson and his AEGIS Math Team, Philippe Fondanaiche, Sudipta Das, Burkart Venzke, Dane Brooke, Lou Cairoli.

The terms of the left side are 1, 4 , 6, 7, 10, 11, 13, 16, 18, 19, 21, 24, 25, 28, 30, 31 while those on the right side are 2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 22, 23, 26, 27, 29, 32.

The algorithm for building the sets is simple:
· Start with the two sets A₁ = {1, 4} and B₁ = {2, 3}.
· The partition of integers from 1 to 8 is A₂ = A₁ È (B₁ + 4), and B₂ = B₁ È (A₁ + 4), where (A₁ + 4) means to add 4 to each of the integers in the set A₁.
· In general, once A_k-1 and B_k-1 have been defined, put A_k = A_k-1 È (B_k-1 + 2^k) and B_k = B_k È (A_k-1+ 2^k).

Proving that this works in general is an interesting exercise in algebra.

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