Mathematics puzzle try it out and give the right explanation for a full RATING?

Question

Part 1.

Six travellers approach a bridge together at night and their respective times for crossing the bridge are 1, 3, 4, 6, 8, and 9 minutes. What is the best way to schedule them to get all of them to the other side in the shortest time possible if there can at most two people per trip and there is only one lantern between them?

Part 2.

Seven travellers approach a bridge together at night and their respective times for crossing the bridge are 1, 2, 6, 7, 8, 9, and 10 minutes. What is the best way to schedule them to get all of them to the other side in the shortest time possible if there can at most three people per trip and there is only one lantern between them?

Discuss both solutions. Provide a convincing argument that your solutions are optimal.

Note:

a) The maximum persons per trip is two in Part 1, and three in Part 2.

b) The slowest person crossing determines the crossing speed for that trip.

c) There is exactly one lamp which must be present with the traveller(s) for the entire duration of every crossing.

d) No memorising, throwing, pyromania or other nonsense is allowed.

Answer 1

answer : q1 = 32 minutes, q2 = 25 minutes

q1 : 1, 3, 4, 6, 8, 9

1, 3 = 3 (3 stayed)

1 = 1 (1 returned)

8, 9 = 9 (8, 9 stayed)

3 = 3 (3 returned)

1, 3 = 3 (3 stayed)

1 = 1 (1 returned)

4, 6 = 6 (4, 6 stayed)

3 = 3 (3 returned)

1, 3 = 3 (1, 3 stayed)

answer = 3+1+9+3+3+1+6+3+3 = 32 minutes

q2 : 1, 2,  6, 7, 8, 9, 10

1, 2 = 2 (2 stayed)

1 = 1 (1 returned)

8, 9, 10 = 10 (8, 9, 10 stayed)

2 = 2 (2 returned)

1, 6, 7 = 7 (6, 7 stayed)

1 = 1 (1 returned)

1, 2 = 2 (1, 2 stayed)

answer = 2+1+10+2+7+1+2 = 25 minutes

in both questions, ppl with closest times travel together, so all the slower ones won't drag the faster others with them. the faster people functioned as "torch / lamp returner" in every return trip. i sent a fast group first, so member of slower groups need not make a return trip.

Answer 2

Part 1

First 1 &  9 ========> Total time =  10 minutes

1 returned ========> Total time = 1 minute

Second 1 & 8 ======> Total time = 9 minutes

1 returned =========> Total time = 1 minute

Third 1 & 6 =========> Total time = 7 minutes

1 returned =========> Total time = 1 minute

Fourth 1 & 4 =======> Total time = 5 minutes

1 returned =========> Total time =  1 minute

Last 1 & 3 ========> Total time = 4 minutes

--------------------------------------...

Total time = 10 + 1 + 9 + 1 + 7 + 1 + 5 + 1 + 4 = 39 minutes

Part 2

First 1, 9 & 10  = 20 minutes

1 returned = 1 minute

Second 1, 7 & 8 = 16 minutes

1 returned = 1 minute

Last 1, 2 & 6 = 9 minutes

Total time = 20 + 1 + 16 + 1 + 9 = 47 minutes

Answer 3

Part 1.

1st Traveler = 1 minute

2nd Traveler = 3 minutes

3rd Traveler = 4 minutes

4th Traveler = 6 minutes

5th Traveler = 8 minutes

6th Traveler = 9 minutes

First the 1st and 6th traveler go across. Time: 9 minutes

          Then the 1st goes back with the lantern. Time: 1 minute

1st and 5th go across.Time: 8 minutes

           1st come back.Time: 1 minute

1st and 4th go across. Time: 6 minutes

           1st come back. Time: 1 minute

1st and 3rd go across. Time: 4 minutes

           1st come back. Time: 1 minute

1st and 2nd go across.Time: 3 minutes

           Done.

Total time: 35 minutes

          

Part 2.

1st Traveler = 1 minute

2nd Traveler = 2 minutes

3rd Traveler = 6 minutes

4th Traveler = 7 minutes

5th Traveler = 8 minutes

6th Traveler = 9 minutes

7th Traveler = 10 minutes

First the 1st, 6th and 7th traveler go across. Time: 10 minutes

          Then the 1st goes back with the lantern. Time: 1 minute

1st, 4th, and 5th go across.Time: 8 minutes

           1st come back.Time: 1 minute

1st, 2nd, and 3rd go across. Time: 6 minutes

           Done

Total time: 26 minutes

I think I did this right. Are the explanations in depth enough?

Answer 4

This is a lot of work for ten points but here goes ! :)

For the first problem, the answer is 34 minutes.

For the second problem, the answer is 26 minutes.

The trips are as follows:

a) 1 minute and 9 minute person travel together - 9 min

b) 1 minute person returns  - 10 min

c) 1 minute person and 8 minute together - 18 min

d) 1 minute person returns - 19 min

e) 1 minute person and 6 minute together - 25 min

f) 1 minute person returns - 26 min

g) 1 minute person and 4 min together - 30 min

h) 1 minute person returns - 31 min

i) 1 minute and 3 min together - 34 minutes

The reason this is optimal is that since the time is necessarily that of the slowest person crossing the bridge, there must always be in one direction the sum of each individual person crossing.  The only thing that can be done to reduce the total time is the time needed to traverse back to pick up another person.   The minimum tme for that is 1 minute, the time of the fastest person.  The reasoning is the same for the second problem.

a) 1 9 10 - 10 minutes

b) 1 - 11 minutes

c) 1 7 8 - 19 minutes

d) 1 - 20 minutes

e) 1 2 6 - 26 minutes