1. Bus service

A fairly simple puzzle from Real Life!

A bus company provides a frequent service for passengers between the bus stations in two major towns.

Each bus journey over the same route between towns takes 35 minutes, following which its driver takes a 5-minute rest break at the bus station, then drives the bus back to the other bus station, and so on, throughout the day.

During the main part of the day, when all necessary buses have been brought into service, the timetable requires that a bus must depart from each bus station for the other town at 10-minute intervals, every 0, 10, 20, 30, 40 and 50 minutes past each hour.

Q1: what is the minimum number of buses required to perform the full service?

Q2: if the rest period after each trip is increased to 15 minutes instead of 5, what is the new minimum number of buses required?

Q3: suppose you were a passenger in the bus on a journey from one town to the other. How many buses would pass you in the other direction during your journey?

I am looking for simple and elegant answers to these questions which an intelligent 12-year-old could understand, not merely a set of numeric answers!

2. Assuming a regular 10 minute service and 35 minute travel time, and assuming no hold ups or driver issues, I reckon

I need some more information for the next question - do drivers stay with their buses? If no then it should be the same (but more than
drivers are required). However this is my answer assuming drivers stay with buses.

q3 You should see

This was done by using Excel (not having any timetabling software). I can attach it if required.

3. Not bad, but for both Q1 and Q2 it doesn't actually matter how many drivers there are available (one per bus is needed, of course, plus they will take much longer breaks at various times of the day).
You can think of the bus as taking the 5- or 15-minute rest break, if that helps!

However, you need to rethink your answer to Q3 bearing in mind the Theory of Relativity...
The result is quite surprising (to me, at least) - assuming I've got it right!

4. So are my answers to q1 and q2 correct? If not, I shall wait with bated breath to see what they are and how I got it wrong. Oh, and I assumed it was the bus that had the break, not the driver - I've seen how long the drivers can take at E* Bus station!

Looking at q3 again, I think my assumptions were wrong - and I think my brain will turn to mush (well more mushy) trying to work it out. When I wrote the answer I wondered if it was too low....

Why aren't our colonial cousins answering?

5. I think the first and second\\\\\\ third spoilers are correct. (Ooops, mistype which I've just noticed!)
For Q3 you need to consider the even flow of buses coming towards you - whenever they started!

6. Q1 = 8
q2 = 10
q3 = 4

7. Correct, except for Q3, I'm afraid!

8. I had to do this diagrammatically, and it looks like it's

My brain hurts

But our American cousins aren't responding...

9. We're here waiting in the wild!

10. Originally Posted by access-mdb
I had to do this diagrammatically, and it looks like it's

My brain hurts

But our American cousins aren't responding...
That's the figure I got!

Perhaps Americans don't have buses any more, and they all travel in luxury limousines driven by chauffeurs?!

I'll post my answers as a PDF later.

11. Here's the answers as I see them. I've had to do them 'by enumeration', so the tables are a bit long in order to show the patterns. Hence the PDF!

12. Very good Batcher! I used a pseudo timetable diagram, times on one axis and stops (two in this case) on the other. Diagonal lines show the buses moving. So one rising from left to right (our bus) and the rest are falling from left to right (the buses we pass). Wherever the lines cross is where the buses pass. This would allow you to incorporate stops as well it would just alter slightly where they pass. Indeed, it could also show what happens when the buses change speed because of speed limits as well.

I would scan the diagram but it's very rough and ready, as well as my scanner being stored because my computer room is being decorated. Won't be long before I'm offline for a while!

13. This does assume that you have not caught the first bus of the day.

cheers, Paul

14. Pedant Paul

15. Originally Posted by Paul T
This does assume that you have not caught the first bus of the day.
As my old teachers used to say, "Read The Question!"!

I point you to my original post, where it says:
During the main part of the day, when all necessary buses have been brought into service...

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