I was recently at an international airport, trying to get from one end of a very long terminal to another. It inspired in me the following simple maths puzzle, which I thought I would share here:

Suppose you are trying to get from one end A of a terminal to the other end B. (For simplicity, assume the terminal is a one-dimensional line segment.) Some portions of the terminal have moving walkways (in both directions); other portions do not. Your walking speed is a constant , but while on a walkway, it is boosted by the speed of the walkway for a net speed of . (Obviously, given a choice, one would only take those walkways that are going in the direction one wishes to travel in.) Your objective is to get from A to B in the shortest time possible.

- Suppose you need to pause for some period of time, say to tie your shoe. Is it more efficient to do so while on a walkway, or off the walkway? Assume the period of time required is the same in both cases.
- Suppose you have a limited amount of energy available to run and increase your speed to a higher quantity (or , if you are on a walkway). Is it more efficient to run while on a walkway, or off the walkway? Assume that the energy expenditure is the same in both cases.
- Do the answers to the above questions change if one takes into account the various effects of special relativity? (This is of course an academic question rather than a practical one. But presumably it should be the time in the airport frame that one wants to minimise, not time in one’s personal frame.)

It is not too difficult to answer these questions on both a rigorous mathematical level and a physically intuitive level, but ideally one should be able to come up with a satisfying mathematical explanation that also corresponds well with one’s intuition.

[*Update*, Dec 11: Hints deleted, as they were based on an incorrect calculation of mine.]

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23 October, 2011 at 10:23 pm

lhmThe optimal strategy of pausing on moving walkways and running between walkways can be seen as a consequence of unit-elasticity. The slope of a tangent to the “isodisplacement” curves gives the time change for a unit change in velocity. this is greater at lower speeds than higher ones due to the convex shape.

In the relativistic case one can draw a similar picture by replacing the velocity v with the rapidity r = atanh v/c. The isodisplacement curves are also convex but are relative to half the rapidity r. To see this let v’->v in the equation from TT’s comment so that dT/dv = -T/v√(1-v²/c²) and solve for r to find T tanh r/2 = const.

25 May, 2012 at 2:15 am

LoriNice puzzle. I realise this is a late response but I didn’t see an intuitive explanation to all three parts without significant algebra manipulations. One possibility is to use a displacement-time (Minkowski) diagram to compare the time taken to cover a given distance at different speeds as shown below (which I hope comes out OK!). Define:

T1(u;v)=net additional time when speed is decreased to u by an amount v

T2(u;v)=net time reduction when speed is increased from u by an amount v

The first diagram shows the Newtonian case in which times in all frames of reference are equal. The three lines T,T’ and T” represent stationary, slower and faster frames of reference respectively. It’s clear that for fixed v, T1(u;v) is increasing in u and T2(u;v) is decreasing in u which implies the answers to parts 1 & 2 respectively.

In the second diagram, time and displacement are now relative to frame of reference and so the X axis label has been omitted. The line segments u and v are orthogonal to the lines T’ or T” for the corresponding times T1′ or T2” respectively (these two cases are actually distinct but have been combined for economy). The T-intercept represents the time dilation for a unit of time in the moving frame T’ or T”. Changing u and holding v fixed in the reference frame T’ or T” implies the slope of the line segments u and v change orthogonal to the lines T’ or T” but the same relationships apply as in the Newtonian case hence the answer to part 3.

25 May, 2012 at 2:21 am

LoriAbove diagram again (some of the symbols were interpreted as html tags):

8 June, 2012 at 7:53 am

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17 July, 2012 at 8:39 pm

Some Miscellaneous Awesomeness « Research in Practice[…] Terry Tao’s airport puzzle. If you have to get from one end of the airport to the other to catch a plane, but you really need to stop for a minute to tie your shoe, is it best to do it while you’re on the moving walkway or not? (I learned this problem from Tim Gowers’ blog.) […]

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22 December, 2012 at 2:40 pm

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3 May, 2013 at 3:40 pm

ybing81Such a model seems to be somewhat similar to the charge transportation in semiconductors, where concepts like “trap” and “mean free path” may find analogues in this ingenious puzzle.

9 September, 2013 at 6:20 pm

anthonyquasI can’t resist mentioning another airport-inspired puzzle that I heard from Isaac Kornfeld. Many airlines have a policy that they will only transport suitcases up to a maximum “linear dimension” (that is height + width + depth). The question is “can one cheat?”

More specifically, do exist suitcases S1 and S2, where the linear dimension of S1 is more than the linear dimension of S2, but still S1 fits completely inside S2 (for example diagonally)???

21 December, 2014 at 8:23 pm

FanAssuming a two dimensional suitcase, the answer is no, basically by repeated use of the triangle inequality.

2 February, 2014 at 8:00 pm

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15 September, 2014 at 7:11 am

Frances WintersClearly the best choice is to avoid the walkways completely. They will be completely blocked by mathematicians tying their shoelaces, cleaning their teeth or whatever.

27 October, 2014 at 4:30 pm

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25 November, 2014 at 7:41 pm

anonymousSimple solution to 2:

Let G be the length of the ground, W the length of the walkways in total.

Let T be the amount of time you can run at the higher speed, v’. Let T_G be the portion of that time running on the ground.

Assume that you spend T time running at speed v’ and don’t complete the journey, otherwise obviously you just run at full speed the whole time.

Then the time to complete the journey is simply:

T + (G-v’ * T_G)/v + w – (v’+u)(T-T_G)/(v+u). This is a linear function of T_G and the coefficient of T_G is:

(v’+u)/(v+u) – v’/v which can easily be verified to be negative because v’ > v. Therefore to minimize the time, T_G should be as large as possible (i.e you should run on the ground as long as you can).

27 December, 2014 at 12:37 pm

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30 July, 2015 at 4:37 am

BasilMoving on platforms provides you with a speed boost(provided the platform is moving your way of course) for as long as the platform lasts.

so it’s an issue of maximising te spent on the platforms therefore your downtime tying your shoelace should be wgile on the platform while spending up your energy to run should happen off the platform since running on the platform would get you less time od the speed boost.

I did happen to have the same thoughts on my last trip to england as well and did in fact postpone tying my shoelace till i got on a platform :) I ran without regard for optimising my route though as they were shouting my name for i was late..

regarding Q3 i’m not familiar with relativity so i cant give an opinion but i’d like to consider how the problem would change given a stamina regeneration factor(although i guess the answer there would be run off platforms, rest on platforms unless youre full rested while on thr platform. also run with a platform within sight to regen stamina on it :p)

22 September, 2015 at 10:03 pm

abhishekthe problem is really simple.its the math that you all invoke is complicated.here a simple rigorous solution:

q1:let the total time taken be T if you dont tie your shoeson your optimal route.let t be the time you take to tie your shoes.

1:if you tie your shoes on ground you take T+t total time.

2:if you tie your shoes on walkway you take T+tv/(u+v).since v/(u+v)<1,therefore T+tv/(u+v)<T+t.therefore tie your shoes on walkway

8 December, 2015 at 12:55 pm

AnonymousI have a similar problem/puzzle that I think would be interesting – can anyone point me to a place where people pose puzzles (and no, it is not a homework problem I am trying to dupe someone into helping me with :) .

It comes from the world of almond harvesting and uncertainty in nut cart positions!

25 January, 2016 at 8:07 am

AnonymousTake your shoe off and run the whole way, using the walkways when possible. Tie your shoe on the plane.