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It’s elementary my dear Watson

Posted by Chris on February 14, 2011 – 7:46 pm

Sherlock Holmes and Dr. Watson were on a motorboat going downstream on the Thames.  They passed a raft at Waterloo Bridge. An hour later they turned back and eventually passed the raft again. The raft was now six miles from Waterloo Bridge.  Sherlock asked Watson, “do you know what the velocity of the river is?”  “That is quite impossible to determine Holmes”, Watson exclaimed.  Holmes replied, “it’s elementary my dear Watson”, and then he explained how easy it was to do it.

So, how did he Sherlock do it, and how fast was the Thames?

(Yes, I know Holmes never used that phrase).


This post is under “Tom” and has 37 respond so far.
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37 Responds so far- Add one»

  1. 1. slavy Said:

    Nice version of an old problem – I like it :) Unfortunately the answer itself gives too many hints, so I will not give it. Actually, even my comment now gives some hints, so I should better stop writing :P

  2. 2. Chris Said:

    Thanks slavy. I hope that this reply hasn’t given a hint either ;) and a LOL.

  3. 3. mike Said:

    isn’t it just 6 miles an hour? …seems too obvious to be right and you know how these riddles go.

  4. 4. Chris Said:

    Hi mike. Try again.

  5. 5. mike Said:

    attempt #2: Ok. So…I’m pretty sure that i spent a bunch of time making the problem harder than it is, but here it goes.

    Assuming they know they turned back after an hour, one of them is carrying a watch at least.

    So speed of River should just be 6mi/(T).

    Where T = total time they spent on the boat trip after they past the boat the first time.

    So it should be something like 1hr + X.

  6. 6. Chris Said:

    Hi mike. Good man for persevering. You are very close. You haven’t made a mistake. You just need to realise one more thing. Then you’ll realise what a doddle the problem really is :)

  7. 7. Karl Sharman Said:

    He looked up at Big Ben….?

  8. 8. SP Said:

    I spent some time on this one and got about as far as Mike. I tried to set up equations with multiple variables but I’m stuck. All I know is that the river speed has to be less than 6 MPH and more than 3 MPH.

  9. 9. Chris Said:

    Hi SP. You and mike are going to feel very silly when the penny drops. You will first kick and then laugh at yourselves.

  10. 10. Karl Sharman Said:

    Is this going to be that the river has no velocity as it is a geographical landmark?

  11. 11. Chris Said:

    This is hilarious. So many clueless – I never would have predicted it. It stumped me for all of 10 to 20 seconds, LOL.

    It is not a silly trick question. i.e. you really can say the speed is v mph where v is a definite number (greater than 0).

  12. 12. slavy Said:

    Maybe it is a time for a hint? Let me try one – you are given the absolute velocity of the boat and you have to find the absolute velocity of the current. However, to do so you need to look at the relative velocity of the boat with respect to the raft (i.e., consider a coordinate axis, centered at the raft, according to which the raft is stationary and determine how the boat behaves with respect to it). I believe this is a strong hint, and hope Chris will not be mad at me!

  13. 13. Chris Said:

    Hi slavy. That’s fine. I think eveyone’s been tortured (nearly) enough. You can explain it later tonight. But let it survive one more evening, thanks.

    In view of the personal hell you guys have been through LOL, if you initially simply state the correct value without the explanation, I’ll give you full credit. I think each of you would like to work it out for yourselves.

    Better still, only check this page when you have the answer or have given up.

  14. 14. Jan Said:

    3 mph
    (didnt look at the hint)

  15. 15. Chris Said:

    Hi Jan. Yahoo, 3 mph is the answer :) .

    Slavy’s hint probably wouldn’t have directly helped (I’m quite sure that was intentionally so).

    I’ll leave the reason until later tonight, even though it’ll only be for posterity.

  16. 16. mike Said:

    @slavvy

    lol, well if you have the velocity of the boat, then the answer is

    (S.1 – S.2)/2 = river speed
    where:
    s.1 = your speed going down river
    s.2 = your speed going up river.

    but yeah, for my answer i just assumed it was just a wooden boat & no way to measure speed.

  17. 17. Chris Said:

    Hi mike, slavy deliberately threw that in so as not to give the entire game away: that’s the reason I emphasised directly in my post 15.

    The question has provided you with all the raw data that you need. You do not need to know how they measured the hour or knew that the raft was 6 miles from the bridge or what their boat’s speed is.

    Jan has given the correct answer. S/he(?) has the same information as you.

    But get your auto-kick, auto-forehead slap and auto-laugh gear warmed up.

  18. 18. Dual Aspect Said:

    Ah!

    I think I get this now that I’ve seen the answer given as 3 mph.

    Without seeing the answer I couldn’t see the logic, but am I right in thinking that it is inevitable that they will have travelled for 1 hour back upstream before they meet the raft again?

  19. 19. Chris Said:

    I’m not sure if I want to blurt out the reason for the answer.

    People, you can vote if you have given up, or if you want a little more time.

  20. 20. Chris Said:

    Hi Dual. Our posts crossed. You are right :) . Now you know why slavy wouldn’t post the answer.

    The cat’s now out of the bag so here’s the dirty details:

    The boat’s speed is with respect to the water. The raft’s speed is 0 mph with respect to the water. So if the boat went on a trip for 1 hour away from the raft, it must have take 1 hour to get back to the the raft, a total time of 2 hours. You might see that more easily if you think of the river as being a still lake.

    The whole river (a moving lake) and all it’s contents have moved 6 miles with respect to the land. So it’s speed must be (6 miles)/(2 hours) = 3 mph.

    Everybody (except Jan and slavy) should now say “Jan and Dual, thank you for putting me out of my misery”. But get over your deep shame first. LOL.

  21. 21. Chris Said:

    If slavy’s hint had been followed, you’d (assuming no goof ups) would have realised that the boat’s speed dropped out of the equation for the river’s speed – so you didn’t actually need to know it.

  22. 22. SP Said:

    I’m not convinced.

    If the river is going 3 MPH then the raft took 2 hours to get 6 miles downstream.

    The boat passed the bridge, went for 1 hour, then turned back. But if it goes for another hour (to match the 2 hours that the raft took), the boat would already be back at the bridge.

  23. 23. Chris Said:

    SP, you must be a landlubber. The boat’s speed is constant with respect to the water, not the land.

    If the boat was doing, say, 5 mph in the water, then it’s land speed would be 8 mph downstrean and 2 mph upstream.

    So after the first hour, it would have gone 8 miles downstream with respect to the land. But the raft would have gone 3 miles (with respect to the land), so they would now be 5 miles from the raft.

    They now go upstream for an hour, and so would move 2 miles upstream with respect to the land. The raft would have travelled a further 3 miles downstream with respect to the land.

    So the raft has moved 3+3 = 6 miles with respect to the land. The boat, also, has moved 8-2 = 6 miles with respect to the land.

  24. 24. SP Said:

    Ok, I understand in your example that the boat goes faster downstream than upstream. However, I assumed it would be a constant speed regardless. Landlubber example: If I drive my car with the wind and it goes 60 MPH, that doesn’t mean it goes less when I drive into the wind. I can still make it go 60 MPH.

  25. 25. Chris Said:

    But it would be different if your car was a plane.

    Your car’s speed is relative to the road, not the air.

    The boat doesn’t use the ground to propel itself.

    I’ve been sailing. I’ve been in currents that are faster than the boat’s speed, so we were going backwards.

    Ship’s often leave at or after high tide, not so much because of the water depth, but so they don’t have to waste fuel fighting the incoming tidal current.

  26. 26. Karl Sharman Said:

    Yep, I’m a landlubber.
    *Slap*
    Now my forehead hurts…

  27. 27. slavy Said:

    Just a clarification: knowing the exact (absolute) value of the velocity of the boat doesn’t help us at all! We have to do the same computations/arguments as here and eventually understand that this number is irrelevant. So mike, the problem becomes even harder with the exact boat speed in the text, because it is a misleading piece of information which most of the people will try to use.

  28. 28. Chris Said:

    Hi slavy, I had taken it for granted that the absolute velocity of the boat was it’s relative velocity plus the river’s speed, so would be r+b downstream and r-b upstream, where r denote’s both the river’s and the raft’s speeds.

    I would then have noted that the boat’s speed wih respect to the raft is b (outwards trip) and -b (return trip) – so Bob’s your uncle.

  29. 29. Chris Said:

    Hi SP. You have mentioned a good point that I hadn’t thought of. The air will have an effect on the original problem. As this is puzzleland, we assume the the wind speed is just right to not affect the solution. i.e. assume there is there is no wind observable from the river, and so the wind speed is 3 mph (downstream) according to a landlubber.

    Even if the wind wasn’t 3mph, the air resistance (force) will be relatively small when compared with the water’s resistance (force). Even that effect will be masked to some extent as the wind will assist in one direction and retard in another. The cancellation isn’t perfect, as has been noted by Dual Aspect in the “the answer is blowing in the wind” problem.

  30. 30. mike Said:

    heh heh…i gotcha now…

    it’s all about getting to that whole ride 1 hour away, ride 1 HOUR back thing that catches you. Not to mention, most the problems on this site require half a page of math work usually O.o.

    either wayy, good riddle : P.

  31. 31. Chris Said:

    Hi SP. This problem and “the answer is blowing in the wind” are causing me to make more observations.

    In the case of your car, the air speed isn’t the overall governing factor – it is more likely to be the statutory speed limit.

  32. 32. Chris Said:

    Hi mike. I’m glad you liked it. It is quite sweet. But I hadn’t imagined that it would get more than 5 or 6 responses.

  33. 33. Chris Said:

    I’ve been Mr. Thickie. Karl mentioned Big Ben and mike mentioned a watch. If they had measured the elapsed time, even Watson could have worked it out.

    Nah, I didn’t catch on, because if that really was the only way to solve it, then the problem would have been a silly trick one. I have only posted one of those in the last two years.

  34. 34. Dual Aspect Said:

    Hi Chris,

    Re your post #31 and SP’s post #24 I think there’s something else fundamental that has been missed.

    I may be wrong but SP states that he can “make his car” go at 60mph regardless of whether it is facing into or with the wind.

    Surely this is because he can adjust the throttle to maintain velocity rather than by any natural physical properties of travelling with or against the wind.

    In the puzzle we are assuming a constant power output for the powered boat. If a car travelled at a constant power output then it would be accordingly slower into the wind that it would be with the wind.

  35. 35. Chris Said:

    Hi Dual. You are right. That’s what I’ve been banging on about, but not very well. Thanks for putting it more clearly.

  36. 36. SP Said:

    Yes I assumed a constant speed for the boat rather than a constant power output.

  37. 37. Chris Said:

    Hi SP. I started writing a long response, but decided that it was counter-productive. [later: but it ended up being long anyway].

    In the case of this problem we have assumed the boat is moving at a constant speed with respect to the water. i.e. We’ve tacitly assumed that there is no wind (according to someone on the raft say) and that the conditions of the river etc., haven’t changed during Holmes’ and Watson’s jaunt. If conditions had changed, I guess that we’ve assumed that the boat’s skipper had tweaked the throttle as appropriate. Welcome to puzzleland, where the first unwritten rule is “keep things as simple as possible, but no simpler than that”.

    In your car, you will adjust the throttle in order to compensate for varying road and wind conditions. If the wind blows against you, you’ll push the throttle down a bit to compensate. If the wind then changes, you’ll change the throttle setting to compensate. You simply aren’t (usually) aware that that’s what’s happening. You simply use your speedometer to decide on what to do. If there was no wind or other variations, you wouldn’t need to adjust the throttle to maintain a given speed. The power provided by the engine, is just right to keep the speed constant. If your car was going faster than the nominal cruising speed for the set throttle, the air resistance would be higher than nominal and so slow the card down, and vice versa. That’s because the air resistance is a function of speed, and it increases with increasing speed.

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