Bridge Crossing with a Torch
I think this is a more famous puzzle than the ones I usually post, but the answer is so satisfying that I thought it deserved a write up.
There are four people trying to cross a rather long bridge at night. The bridge is old and rickety, so only two people can be on it at the same time. Worried about accidentally misstepping, the people are only prepared to walk on the bridge when they, or the person with them, are using the torch. Unfortunately the torch only has enough power for 17 minutes.
The four people all walk at different speeds. One will take 1 minute to cross the bridge, while the others take 2 minutes, 5 minutes and 10 minutes. In order to use the torch effectively, when walking in a group the people will walk at the speed of the slower person.
How can all four people get across the bridge before the battery runs out? Answer below.
No, there isn't a mistake. If your best answer is 19 minutes then you're caught in a line of logic which looks tempting, but is inefficient. Most people jump to the idea that the 1 minute person should be the one to ferry the torch backwards and forwards. So something along the lines of: 10 and 1 head over, then 1 heads back, which takes 11 minutes. Then 5 and 1 head over and again 1 comes back, which has already taken all 17 minutes. Finally 2 and 1 head over, for a total of 19 minutes.
Instead, the key thought to have, is that the two slow people should be dealt with at the same time. However you can't send them over right at the beginning, because one of them would have to head back with the torch. Instead we can send the crack team of 2 and 1 across first, then get either of them to come back. Then 10 and 5 go across and whichever of 2 and 1 is still on the far side comes back. Finally 2 and 1 go across for the second time. Overall this gives a crossing time of 17 minutes.