To get everyone across in 17 minutes, we need to get the slowest people across together; otherwise we are wasting too much time. Once we get them across, how do we not make one of them walk back with the flashlight? Just have one of the faster people already there waiting to sprint the flashlight back across.
person A: 1 minute
person B: 2 minutes
person C: 5 minutes
person D: 10 minutes
1. A & B cross. total time: 2 minutes.
C |==========================| A
D |==========================| B
|==========================| flashlight
2. B comes back. total time: 4 minutes.
C |==========================| A
D | |
B |==========================|
flashlight
3. C & D cross. total time: 14 minutes.
B |==========================| A
| | C
|==========================| D
flashlight
4. A comes back. total time: 15 minutes.
A |==========================| C
B | | D
|==========================|
flashlight
5. A & B cross. total time: 17 minutes.
|==========================| A
| | B
|==========================| C D
flashlight
Another valid solution is to have A bring the flashlight back in step 2.