OEIS A000375: Topswops and the Quest for the Maximum Steps

We explore A000375, the maximum number of topswaps needed to bring the card 1 to the top in any n-card deck under Conway's Topswaps. We explain the simple rules, the termination proof via the Wilf number, and the sharp Fibonacci upper bound φ(n) ≤ F_{n+1} proved by Murray Klamkin. We also cover the Morales–Sudborough quadratic lower bound, the open gap between n^2 and F_{n+1} for n ≥ 20, and the intriguing non-termination of the Topdrops variant. Plus, we touch on computational questions and why this deceptively simple game continues to inspire deep mathematics.

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