Backtracking at the Festival

A complete backtracking run solves the festival — and exposes the wasted work a smarter method will delete.

Fill variables in order Anvil, Brook, Cinder, Dune, trying values 1, 2, 3. Anvil = 1. Brook = 1 breaks the drummer rule; Brook = 2 survives. Cinder = 1 and 2 break the fireworks rule; Cinder = 3 survives. Now Dune: every value must be earlier than Anvil's 1, and nothing is earlier than 1 — three failures, dead end, retreat. Brook = 3, and Dune fails the same three ways. Retreat again: nothing under Anvil = 1 can ever work, and the search proved it by exhaustion. Anvil = 2: Brook = 1 breaks the van rule, Brook = 2 breaks the drummer rule, Brook = 3 survives; Cinder = 3; Dune = 1 — earlier than 2 — and every constraint holds. Solution: Dune at 1, Anvil at 2, Brook and Cinder at 3, one per stage.