## Scooter Ideas

Today I discuss ideas for solving the puzzle I posted previously in my blog: A Scooter Riddle.

Riddle.Alice, Bob, and Charlie are at Alice’s house. They are going to Bob’s house, which is 33 miles away. They have a 2-seat scooter that goes 25 miles per hour with 1 rider, or 20 miles per hour with 2 riders. Each of the 3 friends has a walking pace of 5 miles per hour. What is the fastest time that all three of them can end up at Bob’s house?

Let’s start.

- The scooter and the three people should always be on the move. Moreover, whenever the scooter moves towards Bob’s house, it should carry two people, and only the driver should be on the scooter whenever it is going back.
- We do not care about each individual’s path: whenever people meet at the same place at the same time, we can swap them. Therefore, we can assume that the scooter’s driver is the same person at all times.
- Thinking about it, we can ignore the scooter’s driver and assume that the scooter is self-driving, and there are only two people, Alice and Bob, who need to get to Bob’s house.
- If Alice and Bob arrive at Bob’s house at different times, there is room for improvement: the fastest person should have used the scooter less and walked more, allowing the other person more use of the scooter. Hence, we can assume that they arrive at Bob’s house simultaneously.
- Without loss of generality, we can assume that Alice starts on the scooter first.
- So the scooter should carry Alice for some time, drop her off, come back for Bob, then drop him off, come back for Alice, and so on. Without loss of generality, we can combine all scooter trips by the same person into one ride.
- Thus, the plan is the following. The scooter drives Alice for
*x*hours, drops her off, and she finishes on foot. Meanwhile, Bob starts on foot. The scooter goes back after dropping off Alice, picks up Bob, and drives him*x*hours the rest of the way.

Now this is just algebra. Each person covers 20*x* miles on the scooter and 33 − 20*x* miles on foot. The walking takes 33/5 − 4*x* hours. Thus the total trip for each person takes 33/5 − 3*x* hours.

Now we calculate what the scooter does. The scooter covers 20*x* miles with Alice and the same number of miles with Bob. It covers 40*x* − 33 miles alone. Thus the scooter spends 2*x* + (40*x* − 33)/25 hours in transit. The two times must be the same. So we have an equation: 6.6 − 3*x* = 2*x* + 1.6*x* − 1.32. Thus, *x* = 1.2, and the answer to the puzzle is 3 hours.

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