## The Escalator Puzzle

A man visits a shopping mall almost every day and he walks up an up-going escalator that connects the ground and the first floor. If he walks up the escalator step by step it takes him 16 steps to reach the first floor. One day he doubles his stride length (walks up climbing two steps at a time) and it takes him 12 steps to reach the first floor.

If the escalator stood still, how many steps would there be on sight?

*Thanks to Bilbao for thisoen*

BillyPilgrim| Profile February 5th, 2009 - 10:20 am32

In both cases he walks up a given number of steps and the escalator carries him up an unknown number. However we know that the number of steps the escalator carries him must be half in the second case, since he’s walking twice as fast and thus spends half as much time on the escalator. So:

16+x=y

12*2+x/2=y

16+x=24+x/2

x=16,

y=32 (ans)

cheddarmelt| Profile February 5th, 2009 - 2:20 pmIf we assume that double stride meant double speed (which may not be true), then the escalator has 32 stairs.

Given:

S = “free” stairs (those not climbed)

Within a given length of time:

16 stairs + S = 12×2 stairs + 1/2 S (= total stairs)

-16 -1/2 S -16 -1/2 S

————————————

1/2 S = 8

S = 16

16 climbed stairs + 16 “free” stairs = 32 stairs total.

24 climbed stairs + 8 “free” stairs = 32 stairs total.

Shawn| PUZZLE GRANDMASTER | Profile February 5th, 2009 - 4:58 pm32

hex| PUZZLE MASTER | Profile February 5th, 2009 - 5:19 pm(all speeds are in steps/sec)

v=escalator speed

v1=man’s 1st speed

v2=man’s 2nd speed

d=escalator length in steps

T1=1st time span

T1=2nd time span

v+v1=d/T1

v1=16/T1

v+v2=d/T2

v2=24/T2 (you could be tricked into using 12 instead of 24!!)

Solving the above yields:

v=2*v1

v2=2*v1

Hence d=48 steps

aaronlau| Profile February 5th, 2009 - 9:54 pm(x – 16) / 2 = x – 12 * 2

Steps on sight = x = 32

Assuming same step speed, though physically one may be slower when taking 2 steps at a time.

michaelc| Profile February 5th, 2009 - 10:06 pmI think we need an assumption.

He takes his double strides at the same frequenct he takes his single strides, or in other words he doubles his speed.

Now you have he walks 16 steps in the same time the escalator goes the remaining distance as one equation. The next equation is he walks 24 steps in the same time the escalator goes this remaining distance. These 2 equations are equal.

It’s a bit tricky to solve from there, and I won’t go into the details, but there’s only one variant of the time that satisfies the equations and makes them “make sense”.

The second time is 3/4 of the first. Note the second time must be less than the first as he has in effect doubled his speed while the escalator has remained the same.

And that leaves the 2 equations above both equal to 48 steps.

Escalators are like cool and stuff…

Shawn| PUZZLE GRANDMASTER | Profile February 6th, 2009 - 8:34 amOr would you have to add 1 for the step that you start on at the very bottom, making the total 33?

suineg| PUZZLE MASTER | Profile February 6th, 2009 - 10:12 amthis puzzle for me was very hard, first by intuition I tried and the result was not convincing for me, then I tried to apply logic and maths and here it is my answer(not sure is right just yet but pretty convincing for me at least):

1) the man take 16 steps of length 1(jumps of 1 step)to reach level1

2) the man take 24 steps of length 2( jumps of 2 steps jaja) to reach level 1

So the speed of the escalator remains the same, but you can said the velocity of the man in case 2 is the double that in case 1 following this reasoning here are the equations: (VM= velocity of the man;

VE= Velocity of the escalator; T1= time taken by the man to reach level 1 in case 1; T2= time taken by the man to reach level 1 in case 2)

1)VM*T1=16

2)2VM*T2=24

—> (2)/2(1)–>T2=3/4T1

—> THAT CAN BE TRANSLATED MEANING THAT IN CASE 2 THE ESCALATOR COVER 3/4 OF THE DISTANCE THAT IT COVERS IN CASE 1, BECAUSE THE VELOCITY OF THE ESCALATOR REMAINS THE SAME AND THAT THE MAN COVERS 8 MORE STEPS THAT REPRESENT 1/2 OF THE DISTANCE COVERED FOR HIM IN CASE1 SO 3/2 OF HIS ORIGINAL DISTANCE, SO A RAISE OF 1/2 BY THE MAN REPRESENT A DECREASE OF 1/4 BY THE ESCALATOR, SO I “GUESS” THIS MEANS

THE VELOCITY OF THE ESCALATOR IS THE DOUBLE OF THE VELOCITY OF THE MAN SO IT HAS TO COVER THE DOUBLE OF THE DISTANCE COVERED BY THE MAN

VE*T1= 32 TO REACH LEVEL 1 SO THE ESCALATOR HAS 32 STEPS I GUESS, COOL.

RK| Founder | Profile February 6th, 2009 - 11:22 amHave some of our Big Guns slipped on this one??? Bilbao’s puzzle may be trickier than initially thought :-)

Adam| Profile February 6th, 2009 - 1:33 pm32 steps

Jimmy Anders| PUZZLE MASTER | Profile February 6th, 2009 - 2:55 pmFor clarity’s sake, by a step I mean a stair on the escalator

and a movement of the leg I call a stride. Let x denote the

number of steps the escalator moves in one stride. The first time

the man takes 16 strides. He takes 1 step per stride, and the

escalator moves x steps per stride, so altogether the number of

steps is 16 * [1 + x]. The second time he takes 12 strides at 2

steps per stride and again the escalator moves x steps per stride,

so the number of steps is 12 * [2 + x].

Because there are as many steps on the escalator the first time as

there are the second, it follows that:

16 * [1 + x] = 12 * [2 + x]. Multiplying through:

16 + [16 * x] = 24 + [12 * x], and then rearranging the terms:

4 * x = 8. Now dividing by 4:

x = 2.

So there are 16 * [1 + 2] = 48 steps on the escalator.

Shawn| PUZZLE GRANDMASTER | Profile February 6th, 2009 - 2:58 pmI viewed the problem as follows:

As you walk up the steps, a certain number of steps are disappearing at the top.

x = # of steps that disappear at the top as you walk one-step-at-a-time

y = total # of steps

When you take 16 steps one-at-a-time to reach the top, then x steps disappear, so:

16 + x = y

When you take 12 steps two-at-a-time (for a total of 24 steps risen) to reach the top, then 3/4 (or 12/16) of the amount of steps disappear around the bend (assuming the speed of the man’s stride is identical in both cases):

24 + (12/16)x = y

Solving the 2 equations shows:

x = 32 steps that disappear around the pulley in front of the man as he ascends the escalator

y = 48 total steps showing on the escalator at any given time

bizarette18| PUZZLE MASTER | Profile February 6th, 2009 - 3:00 pm48?

Cute picture.

Ari| Profile February 6th, 2009 - 3:42 pmAssuming the time it takes to traverse the stairs halves when his stride doubles:

16+2s=24+s -> s=8

So the staircase moves at 8 stairs/time it takes to traverse the stairs with the long stride.

Following:

16+2*8 = 24+8 = 32

Thus 32 steps total.

deadskin| Profile February 6th, 2009 - 4:06 pmI would think there would always be the same amount of stairs on sight as the previous day??

If we assume that our man takes one step (stair) a second normally we can say that the rate the stairs move at might be Rs = (Stairs moved per Second). So our man takes 1 stair per second or Rw1 in the first scenario, and Take 16 second to get to the top.

So Rs*16sec + 16stairs = Total Stairs

In the second case he is taking stairs at 2 Stairs per Second or Rw2 and takes 12 seconds to reach the top.

So Rs*12sec + 24stairs = Total Stairs

We can set both equal to one another

Rs*16sec + 16stairs = Rs*12sec + 24stairs

Subtract 16 stairs from both side and Rs*12sec from both side to get

Rs*4sec = 8stairs

ending up with the rate the stairs travel at is Rs = 2stairs per second

So fill that into either equation to show

2(stairs/second)*16(seconds)+ 16(stairs) = 48 Stairs visible on the escalator.

Hope that is right.

suineg| PUZZLE MASTER | Profile February 6th, 2009 - 5:04 pmmy last comment was completely wrong jajajajaja I was assuming that the velocity of the escalator being the same for both cases did not matter, but it matter because the time its not the same and so the distance the escalator advance is not the same also, thinking about it again, thanks RK for the comment, i was not sure when I post my previous message, cool I will think it over again.

suineg| PUZZLE MASTER | Profile February 6th, 2009 - 7:45 pmwell was not that bad my previous comment, I think carefully the problem and the solution is as intuitively I think—> 47 (now i am sure)

All the previous comment is truth, but is missing this D= total lenght of the escalator–> 32(steps covered by the escalator in scenary 1)+ 16(steps covered by the man in scenary 1)- 1( the first step the man takes is also the same step the escalator advance in T1 (of my previous comment)) so 47 my final answer and I am 99% sure never 100%, cool.

diegote| Profile February 6th, 2009 - 11:33 pmTo calculate the speed of escalator I used the escalator displacement

ve*16=TotSteps-16 :1st climb

ve*12=TotSteps-24 :2nd climb

1st term numbers are man steps used as time parameter and 2nd term numbers are escalator steps advanced in each case.

ve=2

Based on “doubles his stride length” v2 = 2v1

Operating with

TotSteps=(v1+ve)*16 – 1st climb

TotSteps=(v2+ve)*12 – 2nd climb

found ve=2v1=v2 and replacing

Total steps = 48.

Migrated| Profile February 7th, 2009 - 1:01 amI am a new person to this site and there are many people who I aspire to, but I have for about 1 year or so, regularly visiting the site and trying to solve various puzzles myself.

Anyway. I think the answer to this logic puzzle is 32.

Ok, so when he walking up one at a time, the escalator is in motion. This can be determined by the last line: If the escalator stood still. From this we can say the escalator is moving. So when he walks up, the escalator also will “cover up” steps as they move in its cycle so we let this number be x. So this is 16 + x (for one step at a time.

When he takes 2 steps at a time, he takes 12 steps, covering 24 steps in total as he skips them, and also since it is in motion, it will also cover steps up as it moves. So we let this new constant be y. It cannot be x as he is moving faster, less steps are covered up. So this can be represented as 24 + y.

So if the escalator is still, 16 + x = 24 + y. And x = 2y because he is moving at double the pace in the second one, so half the steps are covered up when he takes the steps 2 at a time, so the constant in the first must be larger than the second constant at double its value. So by solving the simultaneous equations you get y = 8, and x = 16. Then substitute that back in and you get 32 steps.

Falwan| Profile February 7th, 2009 - 1:23 am32 steps!

Bobo The Bear| PUZZLE MASTER | Profile February 7th, 2009 - 11:38 amI had to make a few assumptions here:

1) Our man takes the same length of time to take a stride, regardless of the length of the stride (questionable), and

2) The phrase “takes him 12 steps to reach the first floor” should be interpreted as “takes him 12 STRIDES to reach the first floor”.

Based on these assumptions, I came up with a total of 48 steps. The first time, it’s 16 steps by manpower and 32 steps by escalator power. The second time, it’s 24 steps by manpower (12 X 2) and 24 steps by escalator power (in 3/4 the time as before).

Aether| Profile February 7th, 2009 - 12:08 pmLets take n to be the number of steps visible on the escalator (ie. the “height” of the escalator). The man needs to cover this number of steps in each case. The escalator is travelling at some rate, v steps per unit time, which is constant in both cases. In each case, the man travels at a rate of x steps per unit time. It takes him time, t, to travel up the escalator.

The algebraic equation for this is: n = t * (x + v)

We must assume that the tame taken for each stride is independent of the length of the stride.

In the first case, the man takes 16 units of time, and travels at a rate of 1 step per unit time, ie. t=16, x=1. Thus:

n = 16 * (1 + v)

In the second case, the man takes 12 units of time, and travels at a rate of 2 steps per unit time, ie. t=12, x =2. Thus:

n = 12 * (2 + v)

Since n is constant in both cases, we can set:

12 * (2 + v) = 16 * (1 + v)

24 + 12v = 16 + 16v

Therefore: 8 = 4v, and v = 2 steps per unit time.

We could plug tis into either equation, but we get:

n = 12 * (2 + 2) = 48,

n = 16 * (1 + 2) = 48

So the height of the escalator is 48 steps, and this is how many steps would be visible if the escalator were standing still…

I think…

suineg| PUZZLE MASTER | Profile February 8th, 2009 - 1:24 pmSomething really weird about this problem jaja; if you add 16 steps of the first case with the 24 steps of case 2 + 8 that is the diffeence between the cases you get 48 steps in both cases you have ths same first step so you subtract it and get 47, the same solution but I think its a lucky shot, because the man double up his speed and the escalator has the double of the man speed in case 1 or the same in case 2 so you get the same result but I think this wont be a good explanation for the answer, cool this was just a funny finding, cool.

chr| Profile February 9th, 2009 - 3:08 amx = speed of elevator relative to the speed of the man. How many “steps” the elevator takes per step he takes.

y = length.

Length = (length traveled by man) + (time spent by man)*(time spent by man)

Normal stride length:

1) 16 + 16x = y

Double stride length:

2) 24 + 12x = y

1) 16x = y – 16

1) x = 1/16y – 1

2) 24 + 12(1/16y – 1) = y

2) 24 + 12/16y – 12 = y

2) y – 0.75y = 12

2) 0.25y = 12

2) y = 48

Length of escalator = 48.

x = 1/16y -1

x = 1/16*48 -1

x = 3-1

x = 2

Number of steps the escalator travels in the time the man uses to travel one step.

michaelc| Profile February 9th, 2009 - 11:29 amA lot of folks got 32 for the answer.

I may be wrong on this, but it looks like the assumption is that since the guy “doubles” his stride and therefore doubles his speed, that the distance is cut in half or the escalator’ part is cut in half.

To visualize why this doesn’t work, think of an ant doubling his speed on the way to New Yor from California, and at the same time, a jet cuts his speed in half. The Jet’s distance doesn’t exactly get cut in half on this problem. The same principle applies to this problem.

Thanks to bilbao for another great problem! Keep up the good work. Where do you get these things from anyway?

RK| Founder | Profile February 9th, 2009 - 12:21 pmBefore the caution was posted, 2 people got the right answer- Hex and MichaelC.

After the caution went up, Chr, Aether, Bobo the Bear, Diegate, Deadskin, Bizarette18, Shawn, and Jimmy Anders got the correct answer too.

Welcome, Migrated

suineg| PUZZLE MASTER | Profile February 9th, 2009 - 1:15 pmI have a question to anyone who got the 48 steps answer; for me when I corrected my first post it gave me 32(steps the escalator advance)+16( steps the man walks) = 48 thats the official answer, however the problem says it took him 16 steps to “reach” the first floor right, so the last step the man makes to reach the first floor( not a step in the stair does not count) thats why I subtract 1 and my answer is 47, Is this reasoning wrong?

suineg| PUZZLE MASTER | Profile February 9th, 2009 - 1:23 pmyeah I prefer 48 now better, cool Hex answer is like my reasoning, what matter is the picture just before the man makes the last step cool, this problem was very cool.

bilbao| Profile February 10th, 2009 - 3:30 amI am pleased you liked this puzzle. It encourages me to post even more difficult puzzles since it seems you are all hard to defeat.

MichaelC, I am professor at Deusto University in Bilbao (Spain) and usually work with puzzles for educational purposes. Whenever any of my colleagues come accross an interesting puzzle they send it to me, and I share it with you all :-)

michaelc| Profile February 10th, 2009 - 9:36 amHi Bilbao,

That’s cool. I had never heard of Bilbao, and therefore had no idea it was the name of a city in Spain. (Geography is another weak subject for me!) I did research it some on Wiki and seen the University of Duesto. Very nice.

That’s incredible, the world is getting smaller all of the time! That’s what’s very cool about the internet and the Smartkit website. I’m an American, and have been to Canada one time, but will probably never travel to any other country. Without a “terminal” such as Smartkit, there would be no way to share such things as “The Escalator Puzzle”!

Here’s to a great site and good puzzles!

Hendy| Profile February 19th, 2009 - 5:36 amIt looks as though assumptions are missing here. It does not explicitly state that the escalator moves at a constant rate. It also does not state that each step taken by the man requires an equal amount of time. One would be very hard pressed to prove this theory on an actual escalator with a real person. If the assumptions are made, then this is simply a theoretical discussion and…”The story, all names, characters and incidents portrayed in this production are ficitious. No identification with actual persons, places, buildings and products is intended or should be inferred.”

But I like the idea anyway…Bravo!