Four monks from Tibet have crossed the border into Nepal, and are trying to reach safety in Dharapani by walking the 24 kilometers on the treacherous Manaslu Trail along the Dudh Khola. They are of differing ages and fitness, and walk at 8, 4, 3 and 2 kilometers per hour, respectively. When several people walk together, they walk at the speed of the slowest person.
Along the way there are 5 hiding places, one every 4 kilometers. These are the only places that they can safely stop to rest and eat. Each hiding place is so small that only one person can occupy it at a time, but a monk can stay there as long as desired. For religious reasons, a monk can enter or leave one of these hiding places, or reach the end of the trail, only on the exact hour. It is unsafe for any monk to stop and wait anywhere other than inside the hiding places or at the ends of the trail.
Because the trail passes through narrow gorges and dense forest it will be dark the entire time they are traveling, and because the trail is so narrow and steep they must use a light to keep from falling off. They have one large torch that two people walking together can use, and they have 9 candles that a single person can use. The torch is lit when they start, and will last 24 hours, but the candles last for only one hour each. The lama, the senior monk, who is the slowest walker, carries the torch and all of the candles. He will give the other monks only one candle at a time, and only at the start or at the hiding places.
What is shortest possible combined transit time for the four monks?
With all the constraints I don't see any way the 3kph monk can do anything with the candles. So I constructed this answer, which gets everyone to the end in 20 hours:
3kph monk stays with the lama at all times.
At the beginning of the trail lama gives 4kph monk and 8kph monk one candle each. (2 candles used)
At 1 hour: 4kph and 8kph reach the first and second hiding places respectively.
At 2 hours: the lama reaches the first hiding place and keeps going. Note that he cannot give 4kph a candle because 8kph is already using the second hiding place.
At 4 hours: the lama reaches the second hiding place and gives 8kph a candle. The lama then backtracks. (3 candles used)
At 5 hours: 8kph reaches the fourth hiding place.
At 6 hours: the lama reaches the first hiding place and gives 4kph a candle. (4 candles used)
At 7 hours: 4kph reaches the second hiding place.
At 8 hours: the lama reaches the second hiding place and gives 4kph a candle. (5 candles used)
At 9 hours: 4kph reaches the third hiding place.
At 10 hours: the lama reaches the third hiding place and keeps going.
At 12 hours: the lama reaches the fourth hiding place and gives 8kph a candle. The lama then backtracks (6 candles used)
At 13 hours: 8kph reaches the end of the trail.
At 14 hours: the lama reaches the third hiding place and gives 4kph a candle. (7 candles used)
At 15 hours: 4kph reaches the fourth hiding place.
At 16 hours: the lama reaches the fourth hiding place and gives 4kph a candle. (8 candles used)
At 17 hours: 4kph reaches the fifth hiding place.
At 18 hours: the lama reaches the fifth hiding place and gives 4kph a candle. (9 candles used)
At 19 hours: 4kph reaches the end of the trail.
At 20 hours: The lama reaches the end of the trail.
Solution
