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A skiing brain teaser

Hello, been lurking for a while.  You all seem like a crowd that might appreciate a brain-teaser.

Two skiers, Jim and Dan are riding up an old, slow chair lift.  Jim says to Dan, "Man, I wish they'd upgrade this lift to a high speed lift".

Dan says "no way, I wouldn't want for this part of the mountain to get tracked out faster after a big storm."

Jim replies "Don't worry!  If you sit at the top of the lift, you'll notice that a chair arrives every 10 seconds, and 4 skiers get off.  But go over to the other side of the mountain where they have the new, high speed lift, ride up that, and sit at the top.  You'll see that a chair arrives every 10 seconds and 4 skiers get off there as well.  So even though the chairs move faster, you have the same number of chairs per minute, thus the same number of skiers per minute, thus the mountain won't get tracked out faster."

Dan thinks for a minute, and replies, "Jim, you're wrong.  If I ride a low speed lift, it takes me 10 minutes to ride to the top, and 10 minutes to ski back down.  But if I ride a high speed lift, it takes me 5 minutes to ride to the top, and 10 minutes to ski back down.  That means that on a low speed lift, I can do 3 runs per hour, but on a high speed lift, I can do 4 runs per hour.  But if I can do 4 runs per hour, than so can everybody else.  And if everybody does 4 runs per hour instead of 3, the mountain gets tracked out faster.  So you see, your reasoning is faulty!"

----

Who is right?  Jim or Dan?  Who's reasoning is faulty, and why?  You may assume that in both the high and low speed scenario, there is some constant number of riders at the mountain (say 200) and the resort has only one lift, the one being discussed here, and that every chair on the lift is always full in both scenarios.

It is the assumptions that are key here.  As long as in either case the chair is always full no matter what kind of backlog or lack of backlog there is at the bottom, then the rate of degradation will be the same.  However, in real life that rarely happens.  What happens is the the slow chair is less utilized than the faster chair because people want to fit more skiing in, so they gravitate to faster chairs.  That impacts which areas of the mountain get tracked out first.

I was told there'd be no math.

The solution is found in the first line of Jim's comment, everything else is just fluff meant to confuse the problem.

Jim says the first lift has 4 skiers arriving every 10 seconds.  He then says the high speed lift on the other side also has 4 skiers arriving every 10 seconds.  The speed of the lift (and everything else) is irrelevant because the number of people actually exiting the lift is the same in the exact same time frame.

4 skiers every 10 seconds on Lift 1 = 4 skiers every 10 seconds on Lift 2 = SAME.

What's my prize?

the answer in your scenerio is that niether lift gets more people up the mountain faster, they both move 2400 people per hour. basically what weighs more a ton of feathers or a ton of lead?

I am disregarding your number because in the real world they do not matter because they are not real world number.

High Speed lift track out mountains faster big olde fat period, end of story and its mostly because they get more people up at less than peak capacity, than a fixed grip.

How stoned are the lifties?  I think that would be the deciding factor...

All other things being equal except for lift speed, the chairs must be more widely spaced on the high speed lift.

Here's the question that matters. DO the lifts cover the same amount of vertical? It is possible that the lifts deliver the same number of people to the top but the hi speed covers more vertical

If Dan's more likely than average to screw up the line for everyone else, then he's right. Giving him the opportunity to get more runs per hour will track the mountain out faster.

The way I read the question, it's the same mountain. The only difference is the lift speed.

Quote:
Originally Posted by lonewolf210

Here's the question that matters. DO the lifts cover the same amount of vertical? It is possible that the lifts deliver the same number of people to the top but the hi speed covers more vertical

If the lifts are both serving the same amount of skiers and every chair is full every time, Dan is wrong.  He won't make it to the top any more often using the high speed lift.  It has less chairs with less butts hanging in the air, so those folks are standing in line at the bottom rather than hanging in the air, but otherwise the delivery happens at the same rate on both sides.  I'm assuming that both lifts are the same length, but it really doesn't make any difference.

If you search a bit you'll find several threads from the past arguing permutations of this problem.

If there are 200 people, 100 on the slopes and 100 on the slow chair, 10 minutes apiece, then there is no lift line.

On the fast chair, there will still be 100 on the slopes, but only 50 on the chair. The other 50

will be waiting 5 minutes in line. It still takes 10 minutes to get up the slopes.

But the pow will get cut up earlier by 5 minutes. So if you're late, there will be 50 more tracks

to deal with. That's my story...

Quote:
Originally Posted by sibhusky

It is the assumptions that are key here.  As long as in either case the chair is always full no matter what kind of backlog or lack of backlog there is at the bottom, then the rate of degradation will be the same.  However, in real life that rarely happens.  What happens is the the slow chair is less utilized than the faster chair because people want to fit more skiing in, so they gravitate to faster chairs.  That impacts which areas of the mountain get tracked out first.

^ This.

In the example given, assuming full capacity, there is no difference in skiers exiting the lift onto the mountain since quads travel three times faster but the chairs are three times further apart.  In real life, there is a huge difference due to the effects of the presence of a quad on the dispersion of traffic around the mountain (preference for a faster chair).

In the example, the high speed is operating at double the speed, but with half the number of chairs as the fixed grip.  The uphill capacity, and therefore skier traffic, is the same (assuming both lifts are operating at full capacity).  If, however, there are no line-ups and the skiers can ski directly onto the chair after each run they will be able to ski more runs on the high-speed.

The chairs are more spaced out on the high speed lift, but the lift moves faster so it takes less time from bottom to top.  Furthermore, high speed chairs slow down when they switch tracks at both the bottom and top of the lift.  That could determine the amount of time between unloads while having very little to do with the actual speed of the chair (the amount of time to go from bottom to top).

Dan is right.

The answer is 1,747.221 rounded to the nearest thousandth.

I didn't read all the responses, so sorry if I'm repeating, but I would say that the amount of ski tracks would be the same.  The variable here, if the same number of skiers unload at the top at the same rate, there is a greater distance between chairs on the high speed lift than the slow lift.

The real answer is, the patrollers would have had time to take more runs to "open" the mountain before the High Speed is opened to the public, thus more tracks.

Quote:
Originally Posted by Posaune

If the lifts are both serving the same amount of skiers and every chair is full every time, Dan is wrong.  He won't make it to the top any more often using the high speed lift.  It has less chairs with less butts hanging in the air, so those folks are standing in line at the bottom rather than hanging in the air, but otherwise the delivery happens at the same rate on both sides.  I'm assuming that both lifts are the same length, but it really doesn't make any difference.

If you search a bit you'll find several threads from the past arguing permutations of this problem.

This is right!  The high speed lift will have a longer line, so Dan's calculation of 10 minutes down, 5 minutes up doesn't hold.  As others pointed out, Jim's reasoning is iron-clad, so it's pretty easy to see that he has to be right.  But it took me a while to figure out why Dan's reasoning was wrong.  It also seemed pretty convincing to me at first.

The interesting thing about this is that apparently, if a lift is running at capacity, and if all else is equal (and of course we know it isn't, but humor me), high speed lifts don't get you to the top of the mountain faster than low speed lifts, since high speed lifts will always have longer lines (again, all else being equal).

so if you get more runs per person on a high speed, that means that with a short line (2 min or less)  or no line, that the hill is (could be) tracked out by a smaller number of skiers making repeated laps more often?  I'm off track. If I can get more laps by spending less time on the chair, and so can everyone else, how are more tracks not being left on the hill, say, per hour.

Quote:
Originally Posted by davluri

I'm off track. If I can get more laps by spending less time on the chair, and so can everyone else, how are more tracks not being left on the hill, say, per hour.

If the crowds for the two chairs are of equal size, you CAN'T get more laps by riding the high speed lift.  Therefore, no more tracks than the fixed gripper.  The same amount of skiers are delivered to the top per hour.

Lines at the bottom are never mentioned, and the question is addressing the number of tracks, not the number of runs one skier gets.  I am right.  Like always.  ;)

Quote:
Originally Posted by Posaune

If the crowds for the two chairs are of equal size, you CAN'T get more laps by riding the high speed lift.  Therefore, no more tracks than the fixed gripper.  The same amount of skiers are delivered to the top per hour.

If you ski right onto the lift each run, you, a specific skier,  will get more laps on a high speed.

more tracks on the hill? OK, so the difference is how many people ride each chair (since the spacing is a timed factor). a six pack will surely get more people up and therefore track out the hill faster than the maxed out fixed double or a detachable quad. that's the preserve the snow issue. how many in each chair.

Quote:
Originally Posted by guroo270

Lines at the bottom are never mentioned, and the question is addressing the number of tracks, not the number of runs one skier gets.  I am right.  Like always.  ;)

Which question are you talking about?  The one that asks, "if I can get more laps...?" or the one that asks who is right, Jim or Dan?

The OP question did not mention lines, it's true, but to answer the question of who is correct lines must be brought into the explanation of the answer. If there were no lines on the high speed lift then the question would be moot.

If davluri's question goes beyond the problem stated in the OP, then that's another story altogether.

sorry, Jim is right.  A pound of feathers still weighs a pound, and if they deliver the same amount of skiers in the same amount of time, the number of tracks is =.

Which one stops more often and which one takes longer to restart?

Who cares?  I just jump off every lift that stops, so the amount of tracks would stay the same.  That's safe, right?

Quote:
Originally Posted by guroo270

Who cares?  I just jump off every lift that stops, so the amount of tracks would stay the same.  That's safe, right?

That's how you poach the roped off best part remaining right?  Seriously, who hasn't dropped a pole or hat for the excuse to duck the roped off stash under the lift?

Quote:
so if you get more runs per person on a high speed, that means that with a short line (2 min or less)  or no line, that the hill is (could be) tracked out by a smaller number of skiers making repeated laps more often?

With no line (lift not running at capacity), a high-speed chair will give more runs/hour than a low-speed one, for both individual riders and overall.  Everyone spends more time skiing and less time sitting on the chair.

If you have two lifts with the same uphill capacity, and both always have all the chairs full, then they both deliver skiers/riders to the top at the same rate.  It's just that with a slow chair you have more people sitting in chairs, and with a fast chair you have more people standing in line at the bottom.

I'm telling you, there is NO DIFFERENCE in the theoretical world between the low speed and the high speed if they are both depositing skiers at the top at the same rate per hour and we don't have skier choices about which chair they want to ride.

Let's assume for a second that the person at the top can only COUNT the number of people getting off the chair and doesn't know what kind of chair it is.  Both are delivering skiers at the same rate.  PERIOD.  Therefore, the trail in question is getting tracked out at the same rate.  Same numbers, same rate of tracking.

The fact that there is not skier choice as part of the equation here just proves how crazy it is to discuss.

Quote:
Originally Posted by sibhusky

I'm telling you, there is NO DIFFERENCE in the theoretical world between the low speed and the high speed if they are both depositing skiers at the top at the same rate per hour and we don't have skier choices about which chair they want to ride.

Let's assume for a second that the person at the top can only COUNT the number of people getting off the chair and doesn't know what kind of chair it is.  Both are delivering skiers at the same rate.  PERIOD.  Therefore, the trail in question is getting tracked out at the same rate.  Same numbers, same rate of tracking.

The fact that there is not skier choice as part of the equation here just proves how crazy it is to discuss.

Jeeze, I never intended anybody to get this worked up about it.  You're right of course, assuming enough riders to saturate either the low or high speed lift.  (If there aren't enough to fill up every single upward-bound chair on the low speed lift, then that lift will have less riders per unit time, since some chairs will necessarily arrive empty.)

Anyway, this wasn't intended to have anything to do with the real world whatsoever.  So-called "brain teasers" rarely do.  Anybody fuming about how artificial my scenario is is both totally correct and totally missing the point.

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