I hadn’t expected this escalator business (and see also here) to go on so long, but there have been a lot of smart comments, and a lot of smart disagreements, and a lot of smart changing and re-changing of minds, some of it the unavoidable consequence of the fact that we might all be using language a little differently.
So here is the geeky (i.e. precise!) version of what I want to say.
I. Your journey consists of some time on the stairs and some time on the escalator. You rest for a total of one minute, which you can take on the stairs or on the escalator (or split it if you like).
II. Define some constants:
W = your walking speed
V = the escalator speed
L = the distance from your starting point to your destination
III. Define some variables:
T_W = the total time you spend walking (perhaps some of it on the stairs, some of it on the escalator)
T_V = the total time you spend on the escalator (perhaps some of it walking and some of it resting)
T = the total time for your entire journey.
IV. Write down some equations:
(1) The total length of your journey (in yards) is W T_W + V T_V, which has to equal L.
(2) The total length of your journey (in minutes) is the minute you spend resting plus the time you spend walking, or 1+T_W, which has to equal T.
In short:
(1′) W T_W + V T_V = L
(2′) 1 + T_W = T
V. Combine (1′) with (2′) to get
(3) T = A – B T_V
where A = 1 + L/W and B = V/W are constants.
VI. Now look at what equation (3) is telling you: Total time for the journey (T) depends ONLY (and in a very simple way) on total time on the escalator (T_V). The more time on the escalator, the shorter the journey. More precisely, an extra minute on the escalator cuts B minutes off your journey — and nothing else can affect the length of your journey.
VII. How does this tie in with your decision about where to rest? Answer: Equation (3) tells you that your decision affects your travel time T ONLY insofar as it affects your escalator time T_V. Once again: An extra minute on the escalator subtracts B minutes from your journey. Period.
VIII. Okay, then, how does your decision about where to rest affect your escalator time T_V? That depends on the model—if we assume a fixed stair length and a fixed escalator length, as in my original example, resting on the escalator gives you more time on the escalator. Therefore it affects the length of your journey. In Bennett’s example, resting on the escalator does not affect your time on the escalator. Therefore it does not cut any time off your journey.
IX. Now have fun: Cook up any crazy example you like. Assume, oh, I don’t know, that resting too long on the escalator causes it to turn into a staircase, or that walking on the stairs will cause them to turn into an escalator, or that the escalator won’t work if you rested on the stairs. It won’t matter. As long as you fix the three constants we started with — your walking speed, the escalator speed, and the total distance you want to cover — equation (3) is going to continue to loom over you, so that your journey time will always be determined entirely by your escalator time, and nothing else can matter.
X. Feel free to generalize! Suppose that instead of resting for a minute, you’re going to sprint for a minute, at some speed T_S — or that you’re going to do some combination of resting and sprinting, or whatever. The analogous equations will still rule the day. Your decisions matter when and only when they affect the amount of time you spend on the escalator, and the benefit of an extra minute on the escalator is always exactly B minutes shaved off your journey, where B is some constant that will depend on the various speeds you assumed.
I still don’t see why it’s so complicated.
You have a two-part journey. You have to walk 8 miles, then drive 6 miles. During either stage, with an equal amount of effort, you can make one extra mile of progress. Which stage do you choose? The walking stage, obviously. Better to cut out one mile of walking than one mile of driving.
You have a two-part journey. You have to stand still on an escalator for 55 steps, then walk up 61 stairs. During either stage, with an equal amount of effort, you can make one extra step of progress. Which stage do you choose? The walking stage, obviously. Better to cut out one step of climbing than one step of standing still.
You always want to cut from the slower, more effortful stage.
In the original question, you stand still on an escalator than stand still on the bottom of the stairs. The analogy isn’t as clear, because if you don’t walk up the stairs, the trip takes infinite time. In other words, getting up the stairs is impossible without steps instead of just hard. But it’s the same thing. A speed of zero is less than the speed of the escalator, which is the rule.
—–
As with the other answers, it doesn’t work if time is fixed instead of distance. You have to walk an hour, then drive an hour. With the same effort, you can add a mile to either stage. Which do you choose? They’re the same, obviously.
Also, the intuition that “you never get anywhere” is actually correct, looked at in this way. “You never get anywhere” is the limiting case of “this stage is slower, so add the distance there.”
I’m guessing this intuitive explanation is what the algebra formalizes.
As Bennett says in comment #18 on the last post “resting on a moving escalator does cover more ground than resting on a stopped escalator”, with “stopped escalator” being taken as a synonym for stairs.
This can be modeled using Steve’s equations as follows
Resting on a moving escalator:
W= 10
V= 10
L= 10
T_W= 0
T_V= 1
T= 1
A= 2
B= 1
Resting on a stopped escalator or stairs:
W= 10
V= 10
L= 10
T_W= 0.5
T_V= 0.5
T= 1.5
A= 2
B= 1
This can be visualized as 2 people standing at the bottom of an escalator that has a start button. One person immediately hits the start button and rides without walking up the escalator. The other person waits (and rests) a minute before pressing the start button and then walks up the escalator. The first person is at the top before the second person has even started ! This has to be because “If you rest on a moving escalator, at least you’re making progress”. This is also completely consistent with the rule that you optimize your travel time by maximizing your time on the escalator.
You can then adjust W,V,T_W and T_V up or down and as long as you make the same adjustments to both versions you get the same relative difference (30 seconds) in travel time.
You can get the results from the original post by adding 14,5 to T_W and 4.5 to T_V (and W and V stay the same) to get the Alice and Bob resultts:
Alice:
W= 10
V= 10
L= 10
T_W= 14.5
T_V= 5.5
T= 15.5
A= 2
B= 1
Bob:
W= 10
V= 10
L= 10
T_W= 15
T_V= 5
T= 16
A= 2
B= 1
But no matter how much complexity you add to the scenario and as long as you add the same complexity to both versions at the heart is the 10 yards escalator ride with a one minute rest time that demonstrates “resting on a moving escalator does cover more ground than resting on a stopped escalator” and that I think shows that “If you rest on a moving escalator, at least you’re making progress” is a good explanation for Alice’s choice being smarter than Bob’s!
Rob: I very much appreciate all your effort. But I’m left with this:
that I think shows that “If you rest on a moving escalator, at least you’re making progress” is a good explanation for Alice’s choice being smarter than Bob’s!
I still think this is a poor explanation for multiple reasons:
1) Unless you add appropriate caveats, it leads you to the wrong answer in Bennett’s example.
2) It does not generalize well to questions like “Is it better to sprint for one minute on the stairs or on the escalator?”
3) It fails to address the key point that bothers so many people: Why is an extra step’s worth of progress when I would otherwise be standing still more valuable than an extra step’s worth of progress when I would otherwise be riding the escalator? A step is a step, after all. (Of my three numbered points, this is the one I originally wanted to emphasize.)
To me at least, the three points above (and maybe one or two more I can list) render your way of thinking quite unhelpful.
Rob: I should add that I do not understand what you’re doing with your numbers. T_W is the speed at which you walk on the stairs. It is a constant. T_V is the speed at which the elevator moves. It’s also a constant. You have T_W switching from 0 to .5 and T_V switching from 1 to .5. But you can’t change the values of these constants within a single example. (If your assumption is that T_W has different values for Bob and for Alice, then of course you’ve stepped outside the setup completely. Obviously it pays to be a faster walker, but we want to keep the walking speed fixed.)
I got from your post:
T_W = the total time you spend walking
T_V = the total time you spend on the escalator
I think that is consistent with my usage ?
(It wouldn’t surprise me if I had messed it up somehow, though)
rob: sorry. my screwup. i read w for t_w and v for t_v. you should ignore me this time.
Reply to points made in #3
1. In the example I give above of a 10 yard escalator journey that can be completed in the specified rest time of one minute, (we can add a 10 yard walk afterwards so that the second person can finish their 30-second escalator ride and catch up a bit) then someone who “rests while moving” will establishes a lead over someone who “rests while not moving” and no amount of additional walking on stairs or escalators after this initial 90 seconds (within the framework of this model) will ever narrow the gap.
If however the escalator journey is specified as not 10 yards but one minute (and the total distance is again increased sufficiently) then this changes. The situation after 90 seconds is identical. Someone who “rests while moving” has the same advantage at that point. But in the next 30-seconds these gains are wiped out as the person walking on the moving-escalator catches up with the person walking on the stopped-escalator (because they are moving faster!).
To (tentatively) generalize this answer (need to think this through a bit more, but I’ll say it anyway) :
There are 2 conflicting things in these models:
A: Resting on a moving-escalator is faster than resting on a non-moving-escalator
B: Walking on on a moving-escalator is faster than walking on a non-moving-escalator
In models with fixed escalator distance optimizing for A always yields betters times (and I see no reasons not to say these better times are due to “resting while moving”). Optimizing for A also increases time on escalator.
In models with fixed escalator time A and B cancel each other out. And escalator time is fixed by definition.
2. Not sure why sprinting is conceptually different from walking, don’t you just bump up the value for W if you want it to be a sprint ?
3. I’m not really sure what there is to be explained here. I am claiming that in the models with a fixed escalator distance that resting on the escalator simply reduces remaining journey time so less steps will be needed – but I see no reason for valuing steps on the moving escalator any differently from steps on the non-moving one. I’m possibly missing something here
Unstated but crucial assumption: What is the person on the escalator and stairs trying to maximize (or minimize)? Travel time? Or Rest time?
When I said
“In models with fixed escalator time A and B cancel each other out. And escalator time is fixed by definition”
I meant
“In models with fixed escalator time optimizing for A or B yield the same result. And escalator time is fixed by definition”
By which I mean if you choose a solution that minimizes time resting while not moving your journey time is the same as if you choose a solution that maximizes time walking on a moving escalator. This is not the case for scenarios where escalator distance is fixed.
So you should even walk backwards down the escalator a little bit.
Oh, I see Steve #3 means sprinting instead of resting not instead of walking!
I haven’t run the numbers but I assume that sprinting on the escalator stage would be a bad idea because it would reduce escalator time ?
I do not think there is any particular intuition to be proven or debunked here so I’m not sure that it would be that interesting to discuss – obviously Steve’s equations could be expanded to deal with it.
The main difference in terms of describing the results would be terminological. Rather than saying things like “Alice is getting off to a good start because at least she (unlike Bob) is resting while moving” you would have to say something like “Alice gets ahead by sprinting on the escalator while Bob merely walks. But now they are both off the escalator and Bob is sprinting while Alice merely walks. He has overtaken her and won the race !”.
The phrase “at least she’s moving while resting” just describes a particular situation (where “resting” not “sprinting” is the variable at play) where one participant is gaining ground by moving while resting at a time the other participant is resting while stationary. In the “fixed distance” version this stage is decisive to the result, in the “fixed time” version it is is matched by a period where the other participant is “walking on the escalator” during the time the first participant is “merely resting while moving”.
In anyone still cares here is my view stated in its simplest form:
– All scenarios involving a fixed distance escalator journey can be reduced down to a single scenario with an escalator ride that can be exactly completed in the specified rest time and a choice of taking the rest either on or off the escalator. I explain this in more detail in #1 above. Resting on thee escalator is the best option.
– This can be explained (as Steve shows) by showing that resting on the escalator buys more time on the escalator. This is definitely a sound explanation. Its actually a very cool way to look at it!
– It can also be explained by the fact that when you rest on the escalator you are at least travelling while you are resting. You could literally see the person resting on the escalator advancing and finishing the journey before the other person has even started in the simplest version of the scenario!
– So these two explanations must both be ways of capturing the reality described in Steve’s equations. The “at least she’s moving” intuition has not been debunked!
– Now Steve raises some great questions in #3 above that I have probably dug all sorts of holes for myself in my attempt to answer. But that does change the fact that the “at least she’s moving” intuition really does have some explanatory power.
@Rob
Here’s another example based on Bennett’s, this time with fixed time and distance:
Suppose there is a 100-yard-long train that can take you 500 yards in one minute. You get on the very back end of the train, and you want in 2 minutes time to be at the point where the front of the train stops (this is a 600 yard total journey). You can walk 100 yards in one minute, and you really want to rest for one minute. Should you rest on the train or after it?
If “travelling while resting” matters, you definitely want to rest on the train, then walk to the front after the train stops. If you rest on the train, after minute 1 you’ve travelled 500 yards (and gotten a 1 minute rest) and after minute 2 you’ve exactly arrived at your destination, with that required 1 minute rest.
But what if you didn’t rest while travelling? Now, you walk the length of the train in the first minute, arriving at the destination at the end of minute 1. In the second minute, you get your one minute rest. So after minute 2, you’ve exactly arrived at your destination, with that required 1 minute rest.
What if you missed the first train, but you know a second train is coming in exactly one minute? Then for the first minute you rest, and for the second minute, you travel on the train AND walk the length of the train. At exactly the end of minute 2, you arrive at the same destination as the other two individuals, having gotten the exact same 1 minute rest they did.
When/where you rested, or how far you travelled while resting, did not matter because it did not increase the amount of time you got to spend on the train. Once again, the “time spent being moved faster” shows that there should be no difference (and there isn’t!), but the prediction that “travelling while moving helps” produced an erroneous conclusion.
@sprobert
Yes, good example.
Its a variation on the fixed time on escalator model and of course as Steve’s equations show, will result in the same journey time whether or not you rest on the train.
I think I’m probably flogging a dead horse here but Let me try and step through why I don’t think it debunks the “at least your travelling” intuition.
– If at the beginning you were give a choice between resting for a minute on the platform or resting for a minute on the train and you wanted to travel as far as could in that minute its obvious what the choice has to be, and (I hope) its obvious that its because its better to “travel while resting”. At the margin, given this choice , you go with “travel while resting”.
– In the model originally described by Steve all that happens after that choice is that additional steps are added “walk up the stairs for 10 minutes”, “walk up the escalator for 4.5 minutes” etc. As these rules are unconditional on your initial choice they leave the benefit from “travelling while resting” during that first minute entirely intact
– In the case of your model, it has implicit conditional rules that specify: “If you chose to rest on the train for a minute , then you have to walk by the track for the next minute, while if you rest on the platform for a minute you get the right to walk on the train”. Because of these implicit rules the benefits accrued from “travelling while resting” in the first minute are rolled-back in the second. But it doesn’t mean they were never there !
– An analogy: Most people have an intuition that its better to be given $20 than $10. And other things equal it is. But if the tax rate on gifts is conditional on gift size: 0% for gifts up to $10.01 but 50% for gifts over $20 the benefit is lost. Does that make the intuition wrong for the “other things equal” case ?
@Rob
“If you chose to rest on the train for a minute , then you have to walk by the track for the next minute, while if you rest on the platform for a minute you get the right to walk on the train”.
Yes. This whole exercise doesn’t make any sense if you don’t want to both (a)rest and (b)walk. If you don’t care about rest, then you walk everywhere as fast as you can. If you can’t walk, then resting isn’t a choice, it’s what you have to do. These aren’t arbitrary rules; they’re fundamental to what makes this problem an intresting choice.
Think of the one-minute rest as a constraint on this problem. We have an old, out-of-shape economist who can’t walk everywhere and needs a one-minute rest. But the economist is allowed to walk for the other minute.
What my example shows is that the when and where of the resting are irrelevant, as long as he gets the same time on the train. Does he walk on the train and then rest? 1 minute of rest and 2 minutes to destination. Does he rest on the train and then walk? 1 minute of rest and 2 minutes to destination. Does he rest before the train and then walk on the second train? 1 minute of rest and 2 minutes to destination. Resting while moving didn’t gain him anything, because it didn’t buy him more time on the train.
“Because of these implicit rules the benefits accrued from “travelling while resting” in the first minute are rolled-back in the second. But it doesn’t mean they were never there !”
If you ignore my third case, then the benefits of the first minute go to the person moving, not resting. The person moving will be 100 yards ahead at the end of minute 1. However, if we require (as a constraint to our problem) a one minute rest, then the person who walked to the front of the train will be resting at the end of 600 yards until the end of the second minute, at the same time that the person who rested on the train arrives there. Resting when moving gained that person nothing, because the person who walked while moving got all the same benefits of the 500 yards of train travel. And the person who walked on the train, by the time his rest is over, is no better or worse than the person who rested on the train. Their choice of when to rest was entirely irrelevant.
@Sprobert
Happy New Year!
Would you recognize that the first minute (where both options involve resting) – taken in isolation – does show that “moving while resting” explains why one option gives a big lead over the other ?
If yes: Then why does the second minute (when neither option involves resting) affect this view ?
if no: Then I suspect we simply have different ideas of what it means for a shorter journey time to be the result of “moving while resting”.
In either case I’d like to acknowledge that Steve has very elegantly captured the essence of the issue in the equations above – and to thank him for allowing me to take up acres of his blog space !
@Rob
Let me spell out my example into three scenarios:
Scenario A: Get on the train immediately, walk the length of the train for one minute, rest at end point for one minute (This fulfills the requirement of getting to the destination and getting one minute’s rest).
Scenario B: Get on the train immediately, rest for one minute during the ride, then walk the remaining length for one minute (This fulfills the requirement of getting to the destination and getting one minute’s rest).
Scenario C: Miss the first train, rest for one minute waiting for the second, then walk with the second train moving for one minute (This fulfills the requirement of getting to the destination and getting one minute’s rest).
You seem fixated on the comparison between the first minute of B and C. And yes, of course, Person from B has a huge advantage after the first minute. That’s because Person from C hasn’t (yet) gotten the benefits of being MOVED. The benefit Person from B got could better be explained by the fact that they were being MOVED. Resting was irrelevant.
This is made clear by the comparison between A and B: the person who doesn’t rest on the train is ahead of both B and C. However, we haven’t yet applied the constraint: the scenario requires a one-minute rest, so once Person A takes his rest, both of the other individuals will catch up to (but not surpass) him. Why? Because all three individuals get the exact same benefit from being moved by train: 500 yards in one minute. How/when/where they rested did not affect this benefit.
“If yes: Then why does the second minute (when neither option involves resting) affect this view ?”
As long as resting for Person C does not mean they can never take a train, then he gets all the same benefits as Person B. It’s like in a baseball game if you ask who is winning after the top half of the first inning: of course it’s never going to be any team but the away team, because the home team hasn’t gotten to bat yet. If Person C gets the same benefit from the second train, then he has already taken his rest at the start of minute two, and he can walk whiling use the train to exactly catch up.
What would allow Person B to maintain their lead over Person C? Well, if there is another train that Person B can get on after the first one, then they would be able to go much farther than Person C. In other words, if Person B’s moving while resting on the first train allows him to get onto a second train (or escalator or moving walkway) that Person C can’t, Person B actually does benefit. And why? Because we’ve increaed the time that Person B spends getting moved by some external force = we’ve proved Steve right again!
I prefer meters. Otherwise brilliant discussion, sorry I missed it.
I played with Steve’s equations a bit to try and specify more clearly what were inputs and what were output.
Version 1 (fixed Escalator length)
===================================
Inputs: W,V,L_W (Elevator Distance), L_E (Stair Distance) , R_M (resting While Moving Time)
Constants: A,B,T_W, T_V,R_N(resting While Not Moving Time)
Output: T
Holding all the Inputs constant except R_M then T varies directly with R_M.
I think this is conclusive proof that Resting While Moving is the single variable that affects total journey time as long as Elevator Journey length is fixed.
Version 2 (fixed Escalator Time)
=================================
Inputs: W,V,T_V ,L, R_M (resting While Moving Time)
Constants: A,B,T_W,L_E (Escalator Distance), L_W (Stair Distance) ,T_V,R_N(resting While Not Moving Time)
Output: T
In this case holding everything constant except R_M makes no difference. Holding everything constant except T_V is the key to changing T.
Conclusions
===========
Looked at this way, the 2 scenarios are not 2 different versions of the same problem but different problems with different solutions,
If you face a situation where you have control over Escalator Time then maximize it to minimize journey time.
If you only have control, over Resting while Moving Time then maximize that to minimize journey time.
Very nice. Thank you for all the discussion.
When I was in high school, I participated in the Math Olympics in Portugal. One of the questions was something like this:
Imagine race that consists in going up and then down the hill. One participant run up at an average speed of 10 miles per hour and then down at 20 miles per hour. Another participant run at a constant speed of 15 miles per hour. Which one arrived first?
The answer, of course, is that the guy with the constant speed wins, but most students argued that they would arrive at the same because the average speed was the same. Of course, they missed the point that the guy with variable speed would spend more time running at 10 than running at 20 and, therefore, his average speed would be under 15.
This escalator problem seems similar. You want to minimize the time you spend on the stairs so that you maximize the average speed.
@Sprobert #17
I think your A ad C are the same but with the rest period (while not moving) either at the beginning or the end ?
I think all I am really doing is observing that if one truncates the end of a fixed time escalator journey to make it fixed length one gets the same results as for a scenario with a fixed length escalator journey. While this is undoubtedly true I can also quite see how it might be deemed a bit of a dodge.