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Heathrow Airport and the Travelator Problem

An everyday puzzle that divides opinion

Yesterday I was at Heathrow Airport, and as I walked through the carpeted terminal and approached the horizontal travelator, I looked down and noticed that my shoelace was undone.  I had left it a bit late getting to the boarding gate so there wasn't time to dawdle.  If I wanted to get to my gate as quickly as possible, what was I to do?

(a) Tie my lace while still on the carpet

(b) Tie my lace when on the travelator

(c) It made no difference where I tied the lace, or

(d) It depended on my walking speed relative to the speed of the travelator.

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This was a real life case of the travelator problem, first described by the mathematician Terence Tao in 2008. It's an engaging puzzle because it involves a situation that most people can identify with, yet for many the mathematical answer is far from obvious.  What, intuitively, do you think the answer is (a), (b), (c) or (d)?

Before tackling it as a puzzle, we have to agree some assumptions:

  1. Your walking speed is not affected by whether your lace is undone or not, an untied lace is just a bit annoying.
  2. You always walk at the same speed, and continue walking when on the travelator (so if you walk at 2m/s and the travelator at 1m/s you are moving at 3m/s when walking on the travelator)
  3. The airport terminal is quiet, so you don't have to worry about getting in the way of other passengers when tying your lace.
  4. It's just as easy to tie your lace on a travelator as on a carpet, and tying a lace takes the same amount of time wherever you do it.

As it happens these assumptions were all reasonably realistic in the case of my Heathrow experience.

Spoiler alert - I'm now going to discuss how people answer this question.

I've found that the most popular answer is (a), that you should tie your lace on the carpet - though the reason given is usually a practical one, that you may as well get it done as quickly as possible, and where there is plenty of space for people to walk past you.  But remember we're in puzzle land here, so we have to stick to the rules and think about the pure problem of getting to the destination as fast as possible.  There are others who come to the same answer for a different reason.  Their hunch is that you need to spend as much time as possible walking on the travelator because that's when you move the fastest.  So if you are going to stop, do it on the carpet.

Those who answer "tie on the travelator" tend to reason that this means you are always moving towards your destination, even when you stop to do your laces. You don't waste any time being stationary.

Supporters of "it makes no difference" say that your journey to the gate involves a fixed time tying your laces and the rest of the time walking at the same speed, so your overall journey time will be the same regardless of where you tie the laces.

And finally supporters of "it depends" reckon that the relative speeds of walking and the travelator are bound to have an impact.  The situation is surely different (they argue)  if the travelator is ten times as fast as your walking speed or if it is ten times slower.

All of these answers might seem plausible.  But only one answer is right.

And the correct answer is..... (b)  To get to the gate as fast as possible you should always tie your laces on the travelator. 

How come? 

The answer can be proved using algebra, but that will only convince mathematicians.

There is a more intuitive way to see why tying laces on the travelator gives you the fastest journey.  Imagine you and your alter ego are walking towards parallel travelators.  You both have an untied shoelace.  Your alter ego stops one millimetre before the travelator and ties the lace.  You meanwhile continue for that extra millimetre, so you are now on the travelator as you start to tie the lace.  To all intents and purposes you and your alter ego start to tie your laces at the same time.  But while alter ego has stopped, you are now being swept along by the travelator.  When the two of you finish tying and simultaneously resume walking you have a lead that you will never lose.

P.S.:  I tied my shoelace on the travelator, and used the extra seconds gained to take a photo of the offending shoe, to the slight bemusement of other passengers.