17 minutes seems impossible as to get 17 minutes you need to have 7 and 10 go at the same time, if they don't go at the same time then you will get more than 17 minutes; 10 and 1 would go this would take 10 minutes, then 1 would go back so he could bring over 7, 7 minutes, the total would be at 18 minutes, then 1 would go back and get 2 and they would both come back, the total is now at 21 minutes. 21 minutes is a good time but it is not the time that we want. 7 and 10 need to go at the same time but if you send them to the other side you would need to send one back to give the lamp to 1 and 2 and this would also make the time longer than 17 minutes. So, we need a way to get both 7 and 10 across at the same time without sending one of them back. The only way to do this would be to send both 1 and 2 first, so that one of them could come back with the lamp and barely use any time. So, we send person 1 and 2 across to the other side, this takes 2 minutes. We need to get the lamp back to the other side so that 7 and 10 can go, so we send 1 back as they only take 1 minute. Our total time is now at 3 minutes. The side with 1, 7, 10 has the lamp and now we can send 7 and 10 across at the same time, now our total time is 13 minutes. This is very good as we are aiming for 17 minutes. Now 1 is left at the opposite side to 2, 7 and 10, who also have the lamp, so we need to retrieve 1. The fastest time to cross the bridge is 2 so we send 2 across to get the lamp to 1. Then 1 and 2 both go back, our total time is 17 minutes as 2 going across to give the lamp to 1 and both 2 and 1 going across totals 4 minutes, which when added to the 13 minute total we had previously, makes 17 minutes. We have achieved our goal!
Summarizing the Bridge Crossings:
1+2 go across. Total time is 2 minutes.
1 goes back with the lamp. Total time is 3 minutes.
7+10 go across. Total time is 13 minutes
2 goes back with the lamp to get 1. Total time is 15 minutes.
1+2 go back across. Total time is 17 minutes.
Now we can experiment with different times. If we have 1, 2, 7 and 8 we can try the same strategy as before. We should get 15 minutes as the 8 is replacing the 10 and the 10 only travelled once so if we use the 8 only once, the time should be decreasing by 2. Let's try it out!
We send 1 and 2 over so that 1 can bring the lamp back for 7 and 8. Our total time is 2 minutes. We send 1 back to give the lamp to 7 and 8 and they both go across. Our total time is now at 11 minutes. We then send 2 across to fetch 1, 13 minutes, then send 1 and 2 back to the other side, our total time is 15 minutes. So we were correct!
This strategy should work for all times where there are 4 different numbers, but what if we have more than 1 of the same number? Let's say we have 4, 4, 5 and 5. If we use the same strategy we will send the 2 quickest people over first, 4 and 4. Our total time is 4 minutes. We then send one of the 4's back over to give the lamp to both 5's. This will take another 4 minutes, our total time is now at 8 minutes. Now we send both of the 5's across, and this takes 5 minutes so our total time is now 13 minutes. We send the quickest time at the other side: 4, back over to fetch the other 4 and then send both 4's back across. This will take 8 minutes in total, so now our total time will be at 21 minutes. Can we get quicker than this? No, because this strategy is the optimal strategy to find the quickest way. If we did 4 and 5, sent 4 back and sent another 4 and 5 and then sent 4 back with the lamp for the 4 that is still there and both if they come back. That is 22 minutes, which is slower even though it's only 1 minute slower.