The Trapped Miners

The Story: A rescue coordinator named Suraj receives an emergency call from an iron ore mine in Jharkhand. A tunnel collapse has trapped six miners underground. They are safe for now but need to reach the surface through a single narrow escape shaft. The shaft can only be climbed by one or two miners at a time — it is too narrow for more. Each miner climbs at a different speed: Miner A takes 1 minute, Miner B takes 3 minutes, Miner C takes 5 minutes, Miner D takes 8 minutes, Miner E takes 9 minutes, and Miner F takes 12 minutes. When two miners climb together, they move at the pace of the slower one. A head torch must be used in the shaft — it is pitch dark. There is only one head torch and it must be passed back down after each trip (it cannot be thrown). The support team above estimates the tunnel will become dangerously unstable in 45 minutes. Suraj grabs his notepad. 'We can get everyone out,' he says to his team. 'Let me show you how.' The Challenge: Six miners with climbing times 1, 3, 5, 8, 9, and 12 minutes must escape a shaft one or two at a time, with one shared head torch that must be returned after each trip. What is the minimum total time to evacuate all six miners, and what is the optimal sequence?