The Bridge and the Torch
The puzzle
Four people need to cross a rickety bridge at night. They have one torch, and the bridge cannot be crossed without it.
At most two people can be on the bridge at once. When two cross together they move at the slower one’s pace, and somebody must carry the torch back for the others.
Alone, the four take 1, 2, 5 and 10 minutes to cross. Get all four across in seventeen minutes.
Stuck?
One at a time, smallest first. Open only as many as you need — the whole point of the puzzle is the part you work out yourself.
Hint 1
The obvious plan - the fastest person ferries the torch back every time - gives 19. Work out why, then look for the waste.
Hint 2
The 10-minute crossing has to happen at some point. Can anything else happen during it?
Show the answer Hide the answer
1 and 2 cross (2 minutes), 1 returns (1), 5 and 10 cross together (10), 2 returns (2), 1 and 2 cross (2). Total: 17 minutes.
How to get there
The natural plan is to let the fastest person shuttle: 1 and 2 over (2), 1 back (1), 1 and 5 over (5), 1 back (1), 1 and 10 over (10). That is 19 minutes, and the reason it is not optimal is that the 5 and the 10 are each paid in full, separately.
Send them together instead. 1 and 2 cross, taking 2 minutes. 1 comes back with the torch, taking 1. Now 5 and 10 cross together, taking 10 minutes - and because they went as a pair, the 5 costs nothing extra; it is absorbed entirely by the 10.
2 is waiting on the far side and brings the torch back, taking 2 minutes. Finally 1 and 2 cross together for 2. The total is 2 + 1 + 10 + 2 + 2 = 17.
The technique worth keeping
Two slow crossings are cheaper as one joint crossing than as two separate ones, because the pair is charged only for the slower. Getting the two slowest onto the bridge at the same time is the whole idea, and it generalises: with more people, keep pairing the slowest together.