On Friday I set this puzzle…..

Yesterday my friend and I started to walk down a road.  We began our walk at the same time, from the same point, and headed in the same direction. I walk at 5 km/h and my friend walks at 6 km/h. Throughout the walk a dog ran between the two of us again and again, with a constant speed of 10 km/h. How far did the dog travel in 1 hour?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

10km, because the dog’s speed is 10 km/h.  Did you solve it?


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. Relieved that I got this one right! Can’t wait for the comments from those who will want to argue that black is white!

  2. I was familiar with the riddle of the pond and the water lillies. The patch of water lillies doubles each day, it takes them a whole week to fill half of the pond, How long will it take them to fill the entire pond? The same kind of thinking -and reading- was required.

    1. I agree, Roland. I asked this same question last Friday. I’d be interested to hear other comments, but I think the answer is, the dog can be anywhere between 5 and 6 km from the starting point. To prove it, just run the scenario backwards. But how can specific initial conditions result in a non-specific result?

      I think this sort of demonstrates the impossibility of the premise:There’s no way for the dog to start going anywhere at 10km/h at the very, very beginning. It has to wait until there is some separation between the two people. It doesn’t have to wait long; in fact it can be as short a time as you like, as long as it’s non-zero. But even very tiny variations in its initial wait time (say 1 microsecond vs. 2 microseconds) can have a drastic effect on its final position.

    2. The classical problem: If two friends separated by 11 kms start walking toward each other at, respectively, 5 and 6 Km/h, and a dog starts running from one to the other (and it doesn’t matter where it is at first, but say it is with friend A*) at 10 Km/h, then we can be sure about where the dog will be after an hour and the distance covered.
      The main difficulty is not the impossibility of an infinite acceleration or deceleration, etc, because even for an abstract problem with massless points moving the way described last Friday, the question cannot be solved.
      Sometimes, you can sum up an infinite series, i.e., the series converges: you must have a formula for every term of the series, and that implies that the first term is the biggest, or at least that they get smaller and smaller in absolute value after a given term. In our case, we don’t have one series: there would be infinitely many but they are not specified.
      Obviously, we can obtain their sum because they all meet a requirement (one hour at 10 Km/h), a piece of information we can use to do it.
      That’s the mathematical origin of Ken’s “non-specific result” even for ideal points moving in unrealistic forms. We know the distance covered by the dog, but Roland’s question cannot be answered.
      Von Neumann’s story ( is based on the possibility to solve the classical problem in two different ways, which doesn’t seem to be the case in the alternative form suggested.

      * which is one of Ken’s remarks

  3. Regarding the questions about the actual path taken by the dog (how many turns? where does it finish after 1 hour?) – these are unanswerable, because with the question as posed, the dog would have to turn an infinite number of times. (If you don’t agree, you need to tell us how many seconds into the walk the dog first turns back from the faster walker).

    To make this work in practice, we’d have to make the dog walk alongside the faster walker for a few seconds, so a gap can open up before it begins the shuttling back and forth. Using a spreadsheet, I came up with the following.

    If the delay before the shuttling starts is 6 seconds, then the dog manages 46 turns. The 46th turn will be 12 seconds before the hour ends. Even-numbered turn, so dog leaves the slower walker and just has time to get ahead of the slower walker by 17m before the hour ends, so 5.017km from the start.

    A small change, using a starting delay of 6.42171 seconds instead means the dog only manages 44 turns, but that 44th turn comes very close to 12 *minutes* before the hour ends. This gives the dog time to almost catch up the faster walker, finishing 4 *millimetres* from the 6km mark (yes, obviously we’re idealing everything has point-sized objects!).

    By fiddling around with that delay, you can achieve any finishing point between 5km and 6km for the dog.

    1. Ken – I think your post complements mine nicely. I also thought about thinking about running the scenario backwards (making it more like the classic fly shuttling between approaching trains). It’s the presence of an infinite number of turns that create the problem of defining the initial conditions uniquely.

  4. I didn’t attempt it as i thought it was a long mathematical problem, and I prefer lateral thinking problems. How foolish of me to not see that it was a lateral thinking problem all along. Good one.

  5. I agree with all the preceding posters about the unreality of this puzzle. A puzzle must be based in reality. It must not require an idealised dog that can (for example) turn around and reverse direction in less than its own body length and in zero time.

    Otherwise the puzzle might as well be “two dog owners fire their dog through a double slit and measure the diffraction pattern (no pun intended)…..”.

    Let us start a Campaign for Reality Based Puzzles.

  6. Thank you for your support @MathMiles. The sooner we have puzzles that are solvable in the real world the better it will be for all of us. Theoretical puzzles do not work in practice.

  7. Barry. Just relax and go with the flow. Less of the OCD and accept the puzzles for what they are – intellectual entertainment and diversion and no more.

    I’m done now

    1. Yet Gabby I am sure we both know that holistic truth and insight comes from toiling away at the detail. This is how and why birth charts work: extreme detail of the precise positions and movements of the entire heavens in order to reach a personal understanding of the destiny of one person.

      If we practice ignoring details we will miss the larger picture that opens up before us.

      A puzzle should help us with our thinking and rational capabilities not hinder us because some random element is considered irrelevant in hind sight.

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