Glasses in the Dark

Yes — win is guaranteed within 5 rounds. Execute the fixed sequence: diagonal flip → adjacent flip → single flip → diagonal flip → single flip. Each step resolves at least one rotation-equivalence class of configurations.

There are only a handful of rotationally distinct ways to arrange 4 glasses in up/down states. The 5-step strategy is a decision procedure that systematically eliminates each non-winning configuration type. Step 1: Round 1 — Flip a diagonal pair: If both were identical, you've made them opposite; if they were already opposite, now they match. This resolves the 'all alternating diagonal' configuration. Step 2: Round 2 — Flip an adjacent pair: Targets the case where two side-by-side glasses are the remaining mismatch. Step 3: Round 3 — Flip any single glass: Converts a 3-up/1-down (or 3-down/1-up) arrangement toward uniformity, or triggers a win directly. Step 4: Round 4 — Flip a diagonal pair again: Catches any remaining rotationally shifted version of the diagonal-mismatch configuration. Step 5: Round 5 — Flip any single glass: This final step is the catch-all for any configuration that survived rounds 1–4. Step 6: Mathematical proof by exhaustion: all 2^4 = 16 starting configurations, when analyzed for their rotation-equivalence classes, collapse to at most 5 distinct types — one per step. Key Insight: Rotation symmetry reduces 16 raw configurations to only 5 equivalence classes. The strategy needs exactly one move to resolve each class. Since each step either wins or reduces the class, 5 steps are sufficient and necessary.