Beaver Math Olympiad Problem 1 (equality of recursive sequences)

Determine whether there exists a positive integer i such that a_i = b_i for the sequences (a_n) and (b_n) defined by (a_1, b_1) = (1, 2) and (a_{n+1}, b_{n+1}) = (a_n − b_n, 4b_n + 2) if a_n ≥ b_n and (a_{n+1}, b_{n+1}) = (2a_n + 1, b_n − a_n) if a_n < b_n; equivalently, decide whether the 6-state, 2-symbol Turing machine with transition table 1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE halts from the all-zero tape.

Background

The authors present a number-theoretic problem that reformulates the halting behavior of a specific 6-state Turing machine. The machine halts exactly when the two recursively defined sequences coincide at some index. Despite compelling probabilistic arguments suggesting nonhalting, the problem remains unresolved.

This problem exemplifies how Busy Beaver investigations generate small, concrete, and challenging open questions; it is described in the paper’s discussion of “Cryptids,” a class of Turing machines with halting behavior believed to be mathematically hard.