The Monk and the Mountain

Let’s do a fun problem.

Consider the following problem stated by Rudolf Arnheim (I’ll give the full reference later):

One morning, exactly at 8 A.M., a monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached the temple precisely at 8 P.M.After several days of fasting and meditation, he began his journey back along the same path, starting at 8A.M. and again walking at varying speeds with many pauses along the way. He reached the bottom at precisely 8 P.M.?

I assert that there is at least one spot along the path the monk occupied at precisely the same time of day on both trips.?

Is my assertion true? How do you decide?

We need all the different ways to think about this on the table, so if you think it is true, how did you decide, and likewise if you think it is false? Make your case as persuasively as you can, and feel empowered knowing that whatever the answer is, those that answer true are as valuable to this exercise as those who answer false.


Thank you to those who participated in the discussion — this is only viable with your input, and I am very grateful. Even with only three responses, you can see that the range of approaches taken is very interesting and useful.

When I first encountered this problem, it was, as I mentioned, in a book by Rudolf Arnheim  More leadingly, the book is Visual Thinking. So I expect that I was primed to pick up a pencil and model the problem. Indeed, my solution was much like Arnheim’s  (just more sketchily), shown below:

Arnhem's visual solution to the monk and hte mountain problem

Arnheim's visual solution to the monk and the mountain problem

What took me by surprise, was that even though we are the “visual architecting” folk, few pick up a pencil to model the problem, preferring to try to solve it with only the benefit of the mind’s eye.  And generally relatively few people in workshops of 12 to 16 very talented people come to a convinced affirmative within the time allotted. What is more, in workshop after workshop, there remained some who were not convinced of an affirmative answer even when I modeled it like this.

So we started doing a simulation along the lines of the thought experiment that Gene Hughson suggests in the comments, except that we would ask one of the participants to help us, and we designated a line across the room as ”the mountain path” and we’d have one of us be the monk going ”up” the mountain and the other being the monk coming down. We’d both start at “8am” and walk in opposite directions, one from the “bottom” (one end of the line) and one from the “top” (the other end). And no matter how fast or slow each of us went, of course there is a point where we bump into each other — when we superimpose the two days, we can see there is a point where we’d be at the same point in the path. Can you envision this in your imagination? Of course, since it is the same path, the entire distance of the path is traversed on both days. (Coming down by way of the same narrow path, we’re always at a point we were on when we went up, just, for the most part, at a different time of day.) The issue is only is there a spot that we’re at, at the same time of day? And if we start at the same time, the answer has to be yes.

Well, it is true on the assumption that what we mean by “same spot” is in the spirit of the problem, so we’re not asking if the monk would step in exactly and precisely the same place (i.e. on the same grains of sand) on both trips. That is, if we don’t complexify the problem. Now, if we are expecting Ruth to try to trip us up with not paying attention to the gtchas in the details, we’d be excused if we anticipated the ways this could go wrong. (Which obviously an architect should also do. In which case we simply need to assert our assumptions, so we state the problem in a way that we can move forward with assurance.)

Still, even with this simulation, some people aren’t convinced. And I love that, because therein lies the most important lesson. Right, we already have important lessons on the table:

  • interpret the problem statement in the spirit in which it is offered, otherwise we over constrain our stakeholders when they are presenting their request to us. This relates well to the gist of Stuart Boardman’s post on Words mentioned in the comments on the Visual Thinking post.
  • get our hands dirty. One way we can do that is by modeling it, as informally as will suffice. Turn it into something we can model visually, mathematically, with a thought experiment or a simulation. To do this, we have to do what we do when we model — we have to use abstractions and representations. We have to decide what is germane and what is unnecessary. Etc.
  • representational redescription: when we present our solution, it is useful to be able to present it different ways. Someone who doesn’t “see it” in one form, may when the information or solution is presented in a different form.

And still there will be some who just don’t see what we may well think is obvious in the light of the solutions presented.  They may come to see it as we present the solution different ways. But they may not. So even when we don’t have entrenched vested interests and other sources of resistance to change to contend with, just at the level of perception there are differences that we have to work with. To work with understanding and empathy, because we are, or could be, those people in the group who just aren’t seeing what other people in the group are seeing.

We need to get modeling and visualization back in the toolbox of our software and technology culture!

27 thoughts on “The Monk and the Mountain

  1. Hi Ruth,

    Based on my sketch at your assertion is true and this was my journey that led to this conclusion:

    I started with a sketch of a mountain with the spiraling path to the temple. Then I sketched the sun which caused the glittering temple. And I sketched the monk and a clock with the finishing time at the temple. And because it was 8 P.M. the sun must be west (based on my own position on earth) so I sketched a little compass just for fun and to take some time to think about how to proceed.
    Next step was to draw some intermediate times along the path to see if I could draw some logical conclusion (pun intended).

    That failed so I started to imagine that I was the Monk:
    “It was a beautiful morning and when I looked up I saw the temple glittering in the sun. I saw myself standing at the end of the path in front of the temple enjoying the amazing view”

    Eureka! Reality is a construction made up by our brain: our mind’s eye. So I (as the Monk) envisioned myself standing before the temple at the end of the first stage (the journey uphill to the temple) looking around and enjoying the view. And a couple of days later I was standing on the same place at this same time before the temple enjoying the view at the start of the downhill journey downhill. And I envisioned myself standing at the end of the path at the bottom….

    Or was it all just a dream within a dream (within a dream)….

  2. Great! Thank you Peter!

    Who else thinks the assertion is true, and how did you convince yourself (now convince us)?

    And who thinks it is false and how did you convince yourself (now convince us)?

    We often do this exercise in workshops and Dana has even done it at a conference, and there have always been people who say true, and others who say false.

  3. My initial reaction on reading this was that it doesn’t seem likely that one could give a categorical answer one way or the other. Given the variable speeds and lack of definition for what constitutes a ‘spot’, it appears that there are scenarios where this could be true, others where it could be false.

    I looked at some solutions that had been posted, but found them lacking. Reformulating the problem to use two monks yields an affirmative, but also introduces a constraint not present in the problem as posed. Having a second monk to block the path forces an answer that I’m not sure would follow with only one monk.

  4. Being a mathematician, my immediate reaction was the same as Gene’s. Both assertions are possible and neither assertion is provable. That’s pretty boring, though and I liked Peter’s story, so I thought about it differently.
    If both assertions are possible and there’s no evidence in the story to suggest a greater likelihood of one than of the other, then I’m free to choose which option I prefer. So I choose true, if only because in this case it’s somehow more romantic than false.

    At this point I considered going into a discourse about quantum mechanics and regarding the monk as a metaphor for an elementary particle. That would allow me to make the bizarre assertion that both true and false are correct (note, not that both possibilities could be true but that both are in fact true). But I won’t.

    So I reverted for a moment to the boring mathematician. Our monk is physical and human and has feet of a finite size and cannot move faster than Usain Bolt. There are therefore only a finite number of places at which true is a possible answer. In fact Peter’s suggestion, that it occurs near the start/end of both journeys, doesn’t seem to belong to the set of possibilities. There are actually more places where false is possible than there are for true. I can draw a picture of that, if you like.

    But we still can’t know for certain and I still prefer allowing ourselves to make a choice. That’s because it resonates with what I find myself saying increasingly often about decisions in what I’ll very broadly call enterprise architecture (but I’ll accept any other kind of architecture, engineering or strategy development). My assertion is that we cannot possibly know whether a decision will prove to be correct. Life is not hard science. We can only make the best possible judgement based on the available information and that judgement will to some extent be influenced by what we would like to be true – our view of the world.
    So I vote true.

    3/15/13: Stuart’s image:

  5. Great discussion, and it raises one of the important points that this problem is so good at surfacing. :-) Ambiguity is so much part of the “air” we breathe as architects, we have to remind ourselves to notice when it is something we can resolve by taking a stand, placing a stake in the ground, as it were, and testing whether we have made the right assumption. In this case, we notice that there is wiggle room in the problem statement, so we can construct arguments like “well, since the path is two feet wide in places; perhaps the monk could happen at those places to step right both times, and so not be at the same place.” Or we can ask ourselves what the intent of the problem is. The problem statement comes from a book by Rudolf Arnheim. Anyone look it up? Yes, it is titled Visual Thinking. Wonderful book, by the way. ;-) Now, if you know that is the title, does that change your approach? Does it change how you think about the problem statement? And how you go about answering it?

    • Funny, I had a very similar picture in my imagination as the one I scrolled down to. Given (as stipulated) that he took the same path down as up, the graphs of elevation vs. time have to cross.

      Of course, it is possible that path could have a big rock in the middle, right at the location where the times of day match, and he might have gone around it to the left in both trips. Maybe this falls into Gene’s “can’t say categorically” category.

      The first response is theory, while the second is practice. As the saying goes, “In theory, theory is the same as practice, but in practice, it isn’t” :-)

  6. I only just saw Arnheim’s solution and at first this led to a bit of an existential crisis. How could I, as a mathematician, have missed this? How could I have been so stupid? I had drawn a picture but it turned out to be the “wrong” picture. Arnheim’s graph (taken on its own terms) is indisputable. But then eureka, I realized that the disagreement between me and Arnheim was because we started from different assumptions – and neither of us included those assumptions explicitly in our pictures.

    I made the assumption that our monk is just an ordinary human being, constrained by the same physical limitations as even Usain Bolt. On that basis, for relatively long distances (such as I hope came across in my picture) there will be a range of places where, were the monk to arrive there at the same time on both journeys, he would be unable to complete either the upward or the downward journey within the 12 hours available in the story. That seems to me to fit with Ruth’s appeal that we should think in “the spirit of the problem”. Implicitly, Arnheim is either not doing that or he is failing to communicate visually the conditions under which his model would be true (i.e. for relatively short distances).
    In case that’s unclear, try shifting the meeting point in Arheim’s graph downwards and to the right. The result is that the monk has to cover by far the largest part of the uphill journey in a very short period of time. This can only be true if either: we constrain the total distance to be short enough or we ignore normal physical constraints. But you can’t read that information out of the graph. Therefore the visual representation is incomplete in one (or more) very important aspects.
    Before anyone jumps down my throat, I’d be the first to admit that my picture also doesn’t make all my assumptions explicit.

    If we choose to ignore the physical constraints of our everyday world, all bets are off and I will let our monk pop in and out of our space/time continuum and therefore meet himself (should he so desire) in three different places simultaneously. That might well require a Feynman diagram to communicate.

    So what conclusions can we draw from this?
    – Perhaps we can’t (always) adequately represent a general solution to a problem with only a picture.
    – Perhaps we can adequately represent the general solution if all relevant assumptions, constraints and pre and post conditions have been stated (in words or via more pictures) or are already known to the reader
    – Perhaps we need to find a way of enriching the semantics of the pictures above in order to adequately communicate all the necessary information – but still be comprehensible.
    – something else

    One thing is clear. Regardless of the limitations from a solution perspective, a picture can still be enormously useful to help us understand the problem and indeed test our own “solutions”.

      • We can argue there is no right and wrong here. Even 1 + 1 = 2 is context specific. I read that an innovative school principal in Russia teaches this with an example along the lines of 1 female guinea pig and 1 male guinea pig pretty soon (and only for a short while!) gets you something like 8 guinea pigs… But 1 + 1 = 2 is often the useful answer.

        Again, the problem framing is important. In positioning the exercise, I allowed that those who answered true and those who answered false were both useful to us — because I didn’t want to influence the solution. I also wanted it to be safe to answer false because those who don’t see the solution provide a highly useful lesson to us too.

        But guess what? My framing still influenced the answer! For the first time we got “both true and false”! Isn’t that interesting? :-) I do see the point that is being made with the “and”. I really do. I also want to be willing to go with the simple answer when it will do. ;-)

        I always learn so much from this exercise; I hope you had fun with it too. :-)

        The biggest lesson to me in all of these discussions is the importance of goodwill — of the willingness to give things a try for the shared benefit of all of us gathered here to learn. That is an amazing grace I deeply appreciate!

      • I first took (like Stuart did, seeing his sketch now) the sketchnoting approach, mainly because I just received (and was primed by) The Sketchnote Handbook by Mike Rohde. And the “problem” with sketchnoting is that you sketch concepts from the original story. And in this case I used those sketched concepts to create my solution sketch & story.

        Now I know I should have followed my tubemapping approach which I normally use first when I try to understand a problem.

        With hindsight I made this image at to show that most information and concept presented by Ruth were irrelevant to solve the problem.

        If you connect two stations with two lines and two trains (one on each line) are travelling in opposite direction with similar arrival and departure times on the same day they always have to pass each other at some point. Regardless how many times they stop (or how fast they go between stops, because the total time will always be the same).

        Other valuable lesson: as an architect you will always be judged with hindsight :-)

  7. Peter, unfortunately you’re absolutely correct in your conclusion. I am indeed now judged with hindsight. :(
    Last night I suddenly realized that I was wrong and also why I was wrong and hoped I might get back in here to announce that before anyone else noticed. Alas, I am too late.

    The bizarre thing is that my drawing illustrated exactly what you have just said but I was so focused on a different problem that I failed to see it.

    I missed the fact that the story makes True be the right answer. We know that the monk completes both journeys in 12 hours, so my objection about physical constraints is ruled out. So, even if that might mean that the monk was superhuman, it’s still a given in the story and I have to live with it. Duh!

    • Hi Stuart,

      I was judging myself in hindsight :-)

      I initially also thought that speed and time were relevant. Most embarrassing thing is that,when I was making my first sketch, I did came to the conclusion that, no matter what, if two monks were doing the same journey in opposite directions during the same period of time they had to meet somewhere. But at that time I thought that wouldn’t be a definitive solution in case that the spot should have the exact size of the footsteps…

      BTW one of my initial solutions was: just take the whole mountain as the spot :-)

    • The monk doesn’t need to complete both journeys in 12 hours. Say, the first monk begins at 8 am and completes his journeys at 8pm going up. The 2nd monk going down just needs to begin his journey some time before 8pm in order for the 1st monk to meet him because obviously the 2nd monk will lie in the path the first monk is taking uphill if starts before 8pm so the 1st monk will have to meet the 2nd monk.

  8. As judging with hindsight goes, our own inner critics can be pretty tough (at least mine are!), but hopefully still much more reasonable than this:
    L’Aquila earthquake scientists sentenced to six years in jail:

    And I thought the lesson was going to be about visual modeling and picking up a pencil — getting “our hands dirty” on the cheap with a pencil because good as our mind’s eye is, seeing relationships out there often allows us to see more (clearly).

    As much as I learn, everyone teaches me I have so much more to learn! It gets uncomfortable doing all that learning in public, so to succor my ego, I make learning the paramount value. ;-)

  9. Perhaps I’m being dense, but showing the paths intersecting is not the same thing as proving the monk must occupy the same spot at the same time of day on two separate days. In this case, I think the visualization is confusing rather than clarifying the issue. No matter how many solutions show that condition, one contrary one will invalidate Arnheim’s assertion.

  10. Interesting problem and even more interesting variations in solution! Here are my reflections on what I did.

    I read the problem online and with no pencil in near proximity sought to solve it mentally. I think I would have been more confident if I could have doodled at a minimum. I like the reference to solving it “in my mind’s eye”, implying that there is a visual element to our thinking, even if we don’t have pen and paper at the time. I certainly created a picture in my mind of what had been described.

    My first picture sought to find an exception – how could they not “cross paths”, so to speak – ie. I eliminated the physical component and started two monks at the same time and asked how could they “not bump into each other”? My mind drew different paths, but then I went back and read that they used the same path.

    But my mind had not realised that I was using “two monks” to solve the problem, so then I created a metaphor or similar problem – I thought of two trains on a train-track. Interestingly, I assumed the track was straight, even though that was not a necessary constraint. Then I worked out that when they crashed would be the time and that this could be anywhere or anytime, depending on the speed of each train. So, this demonstrates how we can use other patterns to solve our problem or to test whether they are the same problem, as well as offer an insight into the problem. Again, I did not explicitly realise that I had eliminated the “different day” element and used “two trains” to solve the problem. (This emerged as I was reading other peoples’ solutions).

    The other two things that “happened” is that I worried about the “at least once” and could not envisage how there could be either no time or more than one time – my train based evaluation convinced me that there is only one time. I also quickly wondered whether I could solve the problem mathematically (since that is my university training of 40 years ago) and decided that was too hard!

    Thanks, Ruth – the “exercise” offers opportunity for lots of different insights!!

    • Thank you Peter! You remind me to mention Paul Zeitz Problem Solving Strategies, and in particular the sections:
      Other Important Strategies 52
      - Draw a Picture! 53
      - Pictures Don’t Help? Recast the Problem in Other Ways! 54
      - Change Your Point of View 58

      • I would like to promote “The Universal Traveler: A Soft-Systems Guide to Creativity, Problem-Solving, and the Process of Reaching Goals” by Don Koberg and Jim Bagnall. Especially p.111 (in the updated classic edition) with “some lessons learned from problem-solving by…experience”, like point 8: Obvious answers are often the hardest to find.

        Which I’ve experienced now :-)

  11. The answer is definitely YES. In fact, we can argue that given the monk starts at 8am from top and bottom, he will have the same distance and same time regardless of whether he finishes the uphill and downhill clim at the same tie or not.

    We can formalize the solution. Let’s consider the monk going up. At any point of time t_i, he would have travelled a certain distance d_i and this is unique since the monk can’t be at two distances in the same time. So, f_uphill: t -> d is defined. Also, f_uphill is continuous since we are covering the entire distance so there’s no break, hence for any d_i there is a t_i for that distance. So, drawing the graph the way the author has drawn makes sense. Same condition applies when coming down (by symmetry).

    Now look at f: t-> d be defined by f(t) = f_uphill – f_downhill. f(0) = 0 – D = -D since initial distance is 0 when going uphill and it’s D when doing down.
    f(t_final) = D – 0 = D. f is also continuous because the difference of two continuous functions is continuous. Hence, by the intermediate value theorem we must attain all the values in [-D,D] at least once for some time t. Hence, we must also attain f(t_j) = 0 -> f_uphill (t_j) = f_downhill (t_j) for this t_j.
    We are getting something stronger here. We now also have that they are not just cross the same point at the same time, but also that they are separated by an arbitrarily close distance for some times t in their walking. :)

  12. Sorry guys there is a mathematical fallacy is the graphing made here. With the way that you are graphing with a single graph, you are asserting that the time on the two days are related to each other. This is absolutely impossible. In order to do this problem you have to make two independent graphs and overlay them on each other and see if the time and the location where they cross match. I can create hundreds of graphs where that is not the case. Just as easily I can create hundreds of graphs where this is the case. There is not enough information to logically make a conclusion one way or the other. Unfortunately I can post my graphs here to display my proofs but if you create the graphs independently and then overlay, you will get the same results.

  13. Without having read all the comments, I am astonished about the diagram. Why are the two curves drawn as mirror images? The way down and up are of course different. Still they will cross somewhere.

  14. The answer is wrong. The fact that the hike took place on two separate days has to be taken into account. If you adjust the intersection of the lines (where the monks cross) on the graph anywhere other than dead center (2 pm in this riddle), then the time for each is different.

    It is true that the monk and his imaginary “shadow monk” will cross each other on the trail at some point if both start walking at the same time, but what the “answer” fails to take into account is that the TIME for both the monk and the “shadow monk” to be in the location is different. Say for instance that the point where they meet is 3/4 of the way up the mountain. On the graph it looks like surely they meet at the same time, but the graph is incomplete. The “shadow monk” on day 1 might make it there at 5:00 pm by his watch by hiking steadily all day. The monk on the second day hiking down at the same pace would be there by 11:00 am by his watch. Yes they would “cross paths” but it certainly would not be at the same time for both because both monks are operating on their own individual timeline unrelated to the each others.

    There is another way to look at it and see that the assumption is wrong. Call the left side of the graph “AM” and the right side of the graph “PM”. Now move the intersection point to the left, to say around what would be 9:00am for both monk and “shadow monk”. It is easy to make the paths cross there on graph, but no matter what height you set the elevation, assuming that it is truly a mountain and not just one step, one monk would really have to hustle to make it to the point on time, and the other monk would have to be leisurely about it. Same if you move the intersection of the graph to the right into the PM area. It is mathematically possible for the monks to cross paths in the same place at the same time, but the probability of is happening is very, very low, and certainly not certain. The assumption stated is incorrect.

  15. More generally, if the starting and ending clock times of the monk’s trek up the mountain are not disjoint from the starting and ending clock times of the next day’s trek down — i.e. if the start-end time intervals overlap — then there will be a same-place-at-the-same-time point for the two walks. For example, the up trek could start at 8 AM and end at 4 PM and the next day’s (faster) down trek could start at 11 AM and end at 2 PM. There will fail to be a same-place-and-time point only if the monk finishes the trek on one of the days before starting the trek on the other day.

    Equivalently, and perhaps easier to “see”, if two monks drive cars along exactly the same route in opposite directions between two towns, and if their start-end travel intervals overlap — i.e. neither finishes the drive before the other starts — then they will pass by each other somewhere along the route at least once, no matter what their speeds are, and no matter if each stops and perhaps even backs up a number of times along the way.

  16. nice – a Zen koan – at first glance appears very difficult to calculate using mathematics – most would throw up their hands – impossible !

    until you relax the mind and just picture it – and the answer is obvious !

    If you imagine the second day as if it were a second person on the same day, one going down while the other is going up – then of COURSE they have to pass each other at some point – the same place at the same time – this is obvious !

    ‘if you meet the buddha on the road, kill him’

Leave a Reply to Ruth Malan Cancel reply

Your email address will not be published. Required fields are marked *


You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>