Please do NOT post your answer, but do say if you think you have solved the puzzle and how long it took. Solution on Monday.

Yesterday I decided to walk from my house to my office.  The journey is 5 miles long and I walked at a steady 2 miles per hour.   When I arrived at the office 40 buses had overtaken me and 50 buses had come from the direction of my office.    What was the average speed of the buses?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.


    1. Lord Manley
      How disappointing, your Lordship. Usually you come up with at least two excellent answers – even if you don’t tell us what they are. We below stairs would be happy to peruse the bus timetables for you…..looks at floor, tugs forelock etc

  1. Surely you can’t tell? They could all have been parked round the corner at the start, whizzed past Richard at 80 mph, and then parked at his office waiting for him to arrive?

  2. I think we have to conclude this problem is set in Switzerland. To solve this it will be necessary to assume the buses travel at uncommonly regular intervals. At least by the standards of the bus services I know.

    1. Fair points, Steve, but I’m more interested to learn Barry Goddard’s theory about the trains in the area. Come on, Bazza!

  3. I don’t see how it is possible to answer this question. Is the bus route only 2.5 miles long – starting and stopping at both ends of your journey? If not, are you walking along a stretch that is a part of one longer bus route. If not, then the buses are from different routes and it is also possible that different bus routes could join the road, travel along it for a short distance and then leave it again.

  4. Despite Stephen’s Jones comment about trains I am in good spirits as my stars are atwinkle today.

    Yes trains would be a better puzzle, Mr Jones.

    When I walk to work I may be passed by a No 30 or a No 86 or a No 253. It depends on what parts of my route I am as my route intersects all those bus routes. Plus a 47 can only pass me in one direction owing due to the One Way System.

    Were I walking from one train station along the track to another and the puzzle were about the trains that pass me in each direction and there are no branchlines to feed in trains onto the track (eg all the trains that pass me either also passed and will pass One station or Another station) then the puzzle is amenable to close thought and an easy solution.

    As stated with buses it is not. I maybe passed by 10 buses whose route only intercepts mine for a half of one mile. Speed then cannot be measured.

    I understand that the usual book solvers of these problems have no problem in disregarding such questions. I prefer to route my solutions in reality for reality cannot be fooled as Richard once said,

    1. Bazza, you’re a gent and a good sport. I can’t work out if you’re an act or if you’re for real, but either way you make me laugh and I thank you for it.

      Steve Jones is blameless in this and you owe him an apology. It was me, not Jonesy, who made the reference to trains, playing on your reputation for introducing irrelevant factors to your solutions. Once again, you have not disappointed.

    2. I was struggling with coming up with a country with the reputation of running their bus services on time. So I just adopted Switzerland as, I imagine, their legendary ability to run trains on time would be transferable onto other forms of public transport.

      I think I have the answer that Richard will come up with. However, I’m not wholly happy with my “proof” in that it’s rather to heavy on algebra and I think there must be a simpler shortcut explanation.

    1. Excellent! The name change was to protect his identity, presumably, and he’s got a few more wrinkles than the last time I saw him at UH, but yes…

  5. Impossible to answer, we do not know enough about the busses.

    He could have been passed by lines 1 through 40 and met lines 41 through 90, with all busses going 80 km pr hour, 50 km pr hour or 79 km pr hour.

    Assuming there is on ly one line, you cannot say anything about the distribution of busses, they may come in clumps of 3 or 4, or he was passed by 30 busses in the first half hour and 10 in the last 5 minutes., You do not know the length of the route, it might be 5 miles or 100 miles.

    The riddle can only be solved if you make a lot of unfounded assumptions, and changing these assumption will give you a new result – as far as I can see.

  6. i think there really are only three assumptions required to answer this puzzle. first, that the buses travel only between wiseman’s house and office (or that bus movements on other routes is neglected). second, that “average” in this puzzle should be read as “mean.”

    and third, that he is walking to the office of the wiseman steamship company.

    1. There are more assumptions required to arrive at an answer. These include buses have to arrive at set intervals, at least at the start and end points (and, of course, that they have to all have the same mean speed between these points). Also, the rate of arrival has to be the same in both directions.

    1. I do not understand the relevance of that as that was last weeks problem.

      But to answer:

      One bus with a fire onboard may create traffic delays in both directions if the road is blocked on both sides while the emergency services are in attendance. Though that may make no practical difference to the problem as stated if the diversionary route is the same for both Richard and the buses in each direction.

      It would mean that he passes at least one fewer bus in one of the directions without affecting any of their average speeds so that would be a difference in the answer, despite the average speed being the same.

      Rather than have a unfruited conversation with an anonymous person concerned about week old problems. Instead I offer a hint for those trying to solve this weeks problem without any extraneous complications.

      Keeping things as simple as Richard and Mr Jones would wish: imagine yet that Richard’s Route to Work is a circle. He does not have to walk the entire circle (that would make his work place the same as his home), He walks a great arc of it.

      Farther imagine the bus route is the exact same as the circle. Buses go round and round all day. Some in one direction (clockwise) and some in the other direction (not clockwise).

      We can now recast the puzzle from the perspective of the bus drivers. How many times on average does a clockwise driver pass Richard compared to a not clockwise diver? We know that answer to be 10.

      The rest is simple.

  7. I found this very hard.
    Took ages to come up with a model and then the (relatively simple) algebra leading to what I believe will be Richard’s answer.
    In that there are no spoilers so far this week I guess that I am not the only person to have difficulty with this.

    1. It certainly has me befuddled, so sorry no spoiler from me this week. Normal service will be resumed next Friday

  8. 6 hours on and off, 5 sheets of finding a model to describe the number of busses, ended up with 2 equations and 2 variables, and got an answer that fits my model.

    It relies on 3 assumptions. I don’t think it’s the solution Richard has in mind, but I’m pretty sure it would be the answer if you were to do it in real life.

  9. It took a few minutes (while I was multi-tasking other things as well) but I assumed that (a) the route from home to office is merely a sub-section of a much longer route which extends beyond it in both directions; (b) the buses go at a steady speed and at regular intervals along the entire route.

  10. Took me about 12 hours counting from the time when I first saw it. . At first I was thinking about it using equations and algebra. But then I decided to get rid of irrelevant information. Then it became simpler and it took me maybe 20-30 seconds.

  11. It is perfectly possible to calculate the number of buses per hour encountered during the walk to work. However, as a now retired bus scheduler, I can assure you that there is insufficient information in the question to calculate the speed of any bus. Route length divided time taken to travel plus layover gives the average speed. Time taken over the whole route divided by desired headway gives the number of buses required.

    1. (I do not consider the following to be a spoiler, but I’m sorry if anyone feels spoiled by it)
      With some assumptions, it’s entirely possible to calculate the speed. You know the frequency of busses passing him from behind and the frequency of busses passing him from ahead. The imbalance between these two frequencies is caused by 1) his walking speed and 2) the speed of the busses. Well, we know the frequencies and his walking speed, that leaves 1 unknown variable that we can isolate and solve for.

      For example: If the busses were slower than walking speed, no busses would ever pass him from behind, but busses would pass him from ahead. Low speed = big imbalance in frequency. On the other hand, if the busses traveled at the speed of light, there would be nearly no imbalance, as they would race by him from both directions. High speed = small imbalance in frequency. All we need to do is calculate the speeds that yield an imbalance of 50 vs 40.

      The assumptions are stuff like “all busses are unique, it’s not 1 bus passing the man again and again”, and “all busses go at the same speed”, and some others.

      Again, sorry if this is a spoiler. I’ll keep my actual answer to myself until tomorrow.

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