The Trapped Miners

All six miners escape in 37 minutes. Optimal sequence: A+B up (3), A down (1), E+F up (12), B down (3), A+B up (3), A down (1), C+D up (8), B down (3), A+B up (3). Total: 37 minutes — 8 minutes before the tunnel becomes dangerous.

With 6 miners, the torch must make 5 upward trips and 4 return trips (11 trips total). The naive approach has the fastest miner (A=1min) escort each person individually, costing 1+3+1+5+1+8+1+9+1+12 = 41 minutes. The optimal approach batches slow pairs. Step 1: Trip 1 up: A+B (3 min). Trip 2 down: A returns (1 min). Elapsed: 4 min. Step 2: Trip 3 up: E+F together (12 min — the two slowest). Trip 4 down: B returns (3 min). Elapsed: 19 min. Step 3: Trip 5 up: A+B (3 min). Trip 6 down: A returns (1 min). Elapsed: 23 min. Step 4: Trip 7 up: C+D together (8 min — next slowest pair). Trip 8 down: B returns (3 min). Elapsed: 34 min. Step 5: Trip 9 up: A+B (3 min). All 6 miners are up. Total elapsed: 37 minutes. Under 45 minutes ✓. Key Insight: The saving over naive (41 min vs 37 min) comes from batching the two slowest pairs (E+F=12, C+D=8) so their slow trips are paid only once each. The cost of extra return trips by fast miners is less than the savings from avoiding individual escort of slow miners.