# When should you rest?

• You are in an airport and are worried about missing your flight.

• So you start to run the long distance to the flight gate at a steady speed of 10km/h (you have bags and are not very fit).

• Luckily some stretches of the journey are covered by moving walkways on which you also run at 10km/h.

• However, you don't have the energy to run all the way and must rest for 1 minute in total at some point before you get to the gate.

• When you rest you walk at 5km/h.

To get there fastest, should you rest on a moving walkway or on some part in between them?

(Please give a full explanation in any answer rather than just a simple guess.)

• Please feel free to suggest better tags if possible. – Lembik Nov 20 '14 at 21:22
• “Luckily some stretches of the journey are covered by moving walkways on which you also run at 10km/h.” 10 km/h in addition to however fast the walkways move? – Ry- Nov 20 '14 at 21:26
• For the record, you shouldn't run for the gate. The best thing to do is to notify the staff on your arrival and they can arrange a transport; infinitelegroom.com/gallery/… – Richard Nov 20 '14 at 22:09
• I freaked out for a minute here; thought you said I run at 10 km/s – Justin Nov 21 '14 at 2:37
• Lateral thinking answer: rest just before the gate, where there is probably a line and you are on sight of the personnel :) – clabacchio Nov 21 '14 at 13:20

$D=$distance to gate

$W=$distance on walkways

$S=$Speed of walkways

$X=$distance travelled while standing

You time to get there assuming you stop off walkway is $$T_r=\frac{D-W-X}{10}+\frac{W}{10+S}+\frac{X}{5}$$ $$X=1/12$$

If you stop on walkway: $$T_w=\frac{D-W}{10}+\frac{W-X}{10+S}+\frac{X}{S+5}$$ $$\frac{X}{S+5}=1/60$$ $$X=S/60+1/12$$ $$T_w=\frac{D-W}{10}+\frac{W-S/60-1/12}{10+S}+1/60$$ $$T_w=\frac{D-W}{10}+\frac{W}{10+S}-\frac{S+5}{600+60S}+1/60$$

$$Tr-Tw=\frac{-1}{120}+\frac{S+5}{600+60S}>0$$

Please note that if $S=0$ then the result is zero.

You should stop on the walkway because you will let the walkway do work to move you during that time.

• Why it works, you spend more time on the walkway where you always move faster. – kaine Nov 20 '14 at 21:54
• You killed the other answer :) – Lembik Nov 20 '14 at 21:57
• Both of them apparently. I wasn't sure I should answer as I wasn't sure it was tricky enough to could as a puzzle... apparently it is... – kaine Nov 20 '14 at 21:59
• +1 for laying out your variables first. I wish more mathematicians would do that instead of expecting people to guess what the letters stand for or wade through text to find out. – Pharap Nov 21 '14 at 1:25
• I think you have the inequality the wrong way round at the end. – Lembik Nov 21 '14 at 21:49

If the distance to the destination is $D$ km and the moving walkway moves at speed $v$ km/h and carries you distance $D_w$ km then:

• If you run all the time on the moving walkway it will take you $\frac{D_w}{v+10}$ hours and if you rest off the moving moving walkway you will do $\frac{1}{60}$ hours resting and $\frac{D - D_w - \frac{5}{60}}{10}$ hours running; and
• If you run all the time off the moving walkway then it will take you $\frac{D-D_w}{10}$ hours and $\frac{D_w - \frac{v+5}{60}}{v+10}$ hours running on the walkway and $\frac{1}{60}$ hours resting on the walkway.

Both options have $\frac{1}{60}$, $\frac{D-D_w}{10}$ and $\frac{D_w}{v+10}$ terms which can be cancelled out so the comparison is between:

• $-\frac{\frac{v+5}{60}}{v+10}=-\frac{1}{60}+\frac{1}{12(v+10)}$ hours if you rest on the walkway; and
• $-\frac{5}{600} = -\frac{1}{120}$ hours if you rest off the walkway.

These two times can only be equal if $v=0$ and if $v>0$ then $-\frac{1}{60}+\frac{1}{12(v+10)} < -\frac{1}{120}$.

Therefore, less time is spent overall if you rest on the moving walkway.

You have to travel distance $d$. At some point, you will stop and rest for one minute, spending time $t$ minutes on a walkway. (If you stopped off of a walkway, $t = 0$. Otherwise, $0 < t \leq 1$.)

During your rest time, you will move $(w + r)t + r(1 - t) = wt + rt + r - rt = wt + r$, where $w$ is the speed of the walkway and $r$ is your running speed.

If you have $W$ metres of walkway out of the distance $d$, without resting, you would cover the distance in $\dfrac{W}{w + r} + \dfrac{d - W}{r}$ minutes. With your rest time, you have $(w + r)t$ less moving walkway, so your non-rest time becomes $\dfrac{W - (w + r)t}{w + r} + \dfrac{d - W}{r} = \dfrac{W}{w + r} + \dfrac{d - W}{r} - t$.

$$\dfrac{W}{w + r} + \dfrac{d - W}{r} - t + 1$$

. Minimizing this means making $t$ as high as possible – 1 – meaning you should spend all your rest time on a moving walkway.

Probably.

• This is the answer. Probably. – Nit Nov 21 '14 at 12:07

No need for any complex calculation!

Let $r$ be your run speed, $w$ your walking speed, and $m$ be the moving walkway speed. If you rest for some time $T$, then you cover the distance $(w+m) \, T$ if you're on the walkway and $w\,T$ otherwise, whereas the distances covered would be $(r+m)T$ and $r\,T$ if running. Let's compare the distance covered by walking for a time $T$ on the walkway and spending the same amount of time running on the unmoving ground, and the distance covered if running on the unmoving ground and walking on the walkway. (I assume that the rest time isn't long enough to cover the whole unmoving ground or the whole walkway.)

• Run on the walkway, walk on unmoving ground: $(r+m) T + w T$
• Walk on the walkway, run on unmoving ground: $(w+m) T + r T$

So whether you choose to rest on the walkway or off it, there is a time period $2T$ (composed of two intervals which may not be consecutive) during which you'll cover the distance $(r+w+m) T$.

Now, how long will it take to cover the rest of the distance? If you run on the walkway, then you'll have less walkway to cover during the time not already accounted for, therefore it will take longer to cover the distance. Hence it's better to run on the walkway.

Illustrated graphically — === is the walkway, --- is unmoving ground, +++ means running, ... means walking, and ←————→ is one of two intervals of time $T$:

• Walk on the walkway:

=============================---------------------------------
++++++++++++++++++++.........+++++++++++++++++++++++++++++++++
←———————→←————→

• Walk on the unmoving ground:

=============================---------------------------------
+++++++++++++++++++++++++++++...++++++++++++++++++++++++++++++
←——————————→←—→


In the second case, more unmoving ground is covered outside the period of time of duration $2T$, therefore the second case requires more time.

Just observe the behavior of crowds, and you get to the right answer: people mostly stand still and rest on escalators, they tend not to do so on staircases. There is a good reason for this.

Key is to realize you have to cover a given distance in a minimum amount of time (rather than maximize the distance covered in a given amount of time). So you have to minimize your slowness (time spend per unit distance), rather than maximize your speed (distance covered per unit time).

For the specific problem posted here: resting in-between moving walkways increases your slowness from $1/10$ to $1/5$ h/km, a net increase of $1/10$ h/km. Denoting the speed of the moving walkways in km/h by $v$, it follows that resting on the walkways increases your slowness from $1/(v+10)$ to $1/(v+5)$ h/km. For any positive $v$ this increase is less than $1/10$ h/km.

Here is another very simple argument. Suppose you are currently 1 meter away from the next moving walkway. Consider two possibilities:

• You may move the 1 meter before resting and move along with the walkway.

• You rest right away, and afterwards move the 1 meter to the walkway.

Both possibilities cost the same time (the time of resting plus the time of walking one meter). Clearly, the first possibility brings you farther ahead.

Hence without any calculation: You should rest on the moving walkway.

• note that the problem statement implies "resting" to mean "walking a full minute at reduced speed". – Johannes Jan 26 '15 at 11:49

Not sure if this is right after seeing the other answers, but I think it doesn't matter if you rest on the moving walkway or not (maybe someone can correct me?)

EDIT: Apparently it does matter when you choose to walk. It seems walking on the Moving Walkway is the better option.

In my previous example I was only calculating with time and not distances. Using my previous reasoning and adding distance this time, my calculations are as followed:

I'll assume the moving walkway moves at 10km/h (same as run speed). This leaves us with 4 speeds:

• Run without MW: 166.7 M/Min
• Walk without MW: 83.3 M/Min
• Run with MW: 333.3 M/Min
• Walk with MW: 250 M/Min

(MW = moving walkway)(numbers have been rounded)

Let's also assume we have to walk 1km, of which the last half is a Moving Walkway (so 500m normal walkway, 500m moving walkway).

If we choose to walk on the normal walkway the whole trip will take us:

• 1 minute walking
• 2.5 minutes running
• 1.5 minutes running (on the moving walkway)

Total: 5 minutes

If we choose to walk on the moving walkway the whole trip will take us:

• 3 minutes running
• 1 minute walking (on the moving walkway)
• 0.75 minutes running (on the moving walkway)

Total: 4.75 minutes

This difference can be explained by looking at the difference in speeds in ratio. The difference in walking/running on the normal walkway is 1:2, which means you are 50% slower when walking. On the moving walkway the ratio is 3:4, so you're only 25% slower.

The amount of time you save depends on the speed of the moving walkway. The greater the speed, the more time you save if you rest on it.

I could try to add a calculation here but I'd probably end up copying the other answers :)

My previous (faulty) reasoning:

We have 4 different speeds (I changed to meters/minute for easy calculation):

• Run without MW: 166.6667 M/Min
• Walk without MW: 83.3333 M/Min
• Run with MW: 333.3333 M/Min
• Walk with MW: 250 M/Min

(MW = moving walkway)

Only two minutes of time are relevant (all other time will be spent running):

• One minute without the moving walkway
• One minute with the moving walkway

This gives us two options

A. Run the first minute, walk the second (on the MW)
B. Walk the first minute, run the second (on the MW)

Option A gives us 166.6667 + 250 = 416.6667 meters
Option B gives us 83.3333 + 333.3333 = 416.6666 meters
(I think we can neglect the minor difference, which is probably caused by rounding)

A different way to look at it, is that the moving walkway will always move you 166.6667 meters per minute extra, regardless if you walk or not. If you split this value from the rest in the calculation, it becomes more obvious:
Option A gives us 166.6667 + (166.6667 + 83.3333)
Option B gives us (166.6667 + 166.6667) + 83.3333

• your flaw is in "Only two minutes of time are relevant (all other time will be spent running)" since if it were to make a difference, one of the two approaches would have less overall minutes. The distance is the same in both cases but you can't assume the time is the same and then use that to prove the time is the same. Also, you have not been told the speed of the walkway. – Kate Gregory Nov 21 '14 at 12:52
• Ah you're right. I misread the part of the Moving Walkway speed (it's run speed apparently). You're also right about the distance I didn't take into account. I'll extend my answer in a few minutes. Thanks for pointing it out :) – Deruijter Nov 21 '14 at 12:58

Lets say we have to rest 1 hour instead of 1 min for easier calculation. Speed increase is linear if you run\walk on moving walkways(MV). Distance you will cover on MV: 1)run (10 + 10) * 1 = 20 km 2) rest (10 + 5) * 1 = 15 km Distance you will cover on road: 1) run 10 * 1 = 10 km 2) rest 5 * 1 = 5 km What we see here is that difference covered distance is constant to 5km, so the answer is doesn't matter, take a rest whenever you want.

UPD: Another way of thinking. We can translate to new coordinate system, where all speed is decreased by 5km\h. This means we run 5km\h and MV is also 5km\h and now rest means stop moving. Obviously we get the same answer - doesn't matter.

• I'm afraid not. You're correct that there is a running time that compensates the walking time either way, and the distance covered during the walking time plus compensating time is the same. But while the remaining distance is therefore the same either way, it isn't split in the same way between the walkway and the steady ground. It's better to leave more walkway for the rest of the time. – Gilles 'SO- stop being evil' Nov 22 '14 at 1:38