Ski Lift Mathematics (February 2024 Challenge Question)

180

His "ascent" is all the way to the top.

I'm equal parts proud and ashamed you got this so quickly.

Nonetheless - congrats!

Look at the diagram in the #27 answer. You do not pass the chair in front of you and you do not pass the chair behind you because they are on the end when you get off and on the chair lift. You also do not pass the chair you are on. That makes the correct answer 177

I think you are confusing ascent with summit...

I think I am using the word ascent like the noun trip.

If you go to A to B via C, your trip is from A to B: especially if the passing of point C is only noticed coincidentally by a third party.

Chair number 100 must have been at the top when he boarded at the bottom for him to pass it half way up. When he started, #10 - #99 were going up, 90 cars, and 90 cars going down. So he passed 90.

44....

177

I had to diagram it.

At the halfway point including 10 and 100 there are 91 chairs to the top total.

That means 45 on each side and one on the end turning around. Because there is one turning around at each end this means there will be two at least two chairs he does not pass. so 180 minus the two on the end turning around and minus the one he is sitting in.

For the chair 100 and the chair 10 to meet exactly halfway on the skilift, chair 100 must be stopped at the very top while Mark Frank is getting on board on chair 10, which means there are about 98 chairs in between them..

To make the loop, the 98 /2 = means that Frank riding #10 on the way up must pass at most 44 or 45 chairs in order to encounter and/or meet chair #100 midway up the skilift.

98?

I think you meant 90.. after all 98/2 doesn't equal 44 or 45

My response to the original question: Too many assumptions within the original question. Equal spacing is good info, but not enough. Chairs (with their numbers) are often removed for maintenance and moved around in position on the lift cabling. No reason to assume the numbers have remained in sequence on the lift.

Yes you are right and it was a mistake on my part.

Thank you for clarifying and pointing out this error in my thinking….

Since you are starting at the bottom of the lift in car #10 and at the halfway point to the top you pass #100, that must mean there are 90 cars coming and 90 going, since we are only passing cars on the adjacent side and only passing half the cars that would mean we passed 44 cars at the halfway point since you can't pass yourself...neither are you passing cars that are travelling with you, the only cars you are passing are going the opposite direction...when you get to the top you will have passed 89 cars in total..but we don't have any information saying we made it to the top, only that we passed car #100 halfway up...at least that's my take...

I concur and shares your analysis...

90 cars going up (#10-#99) and 90 going down (#100-#180, #1-#9) - correct

If the cars going down were not moving, you would pass 90 cars on your trip up.

But as you are going up, the cars on the other side are going down. At the halfway point, you pass #100, the top car coming down when your started. For the rest of your trip you are passing the cars on the way down that were in front of you when you started.

90 of the original cars going down + the 89 cars in front of you = 179 cars passed

I think to enable the meeting of the two chairs, #10 and #100 exactly halfway in their travel, chair #10 should be stopped at the bottom to enable Frank to get on and board the chair #10.. Now in order for the two chairs to meet halfway, chair #100 must also be stopped at the top end of the skilift... which means that the length of the trip must be divided into 2 equal parts..

Yes, my mental picture is one car at the bottom and one car at the top where, at each, passengers exit and enter. Between these two cars are two rows of 89 cars, one ascending and the other descending. Think of a very vertically elongated, 180 sided polygon, with the entry/exit vertices at the top and bottom.

When #10 is at the bottom, #100 is at the top, and, by symmetry, #10 passes #100 at the halfway point.

When ascending, #10 passes the 89 cars in front of #100, #100, and the 89 cars in front of himself.

I'm sorry, I got lost when you added one additional row to make the 2 rows ....

I was under the impression that the ski lift is just one continuous string of chairs.. with chair #100 serving as the lead... and the 2nd loop was created when chair#100 started its descent to looped back downward simultaneously pulling chair#10 to start its accent..

It is very possible for the skilift to have strung only 2 chairs, one on each end of the skilift trip that arbitrarily were labeled as chair#10 on the opposite end of the loop that was labeled as chair #100...

As it is also very possible for the skilift to have a finite number chairs to be equally spaced along one continuous string that was looping around.. In which case the chairs can be sequentially labeled as chair#..... #s... The limiting factor would only be the length of the looped string!

Ah yes of course relative speed, if you are travelling at 3mph and they are travelling at 3mph in the opposite direction then your relative speed would be 6mph, and doubling the number of chairs passed....then at the halfway point it would be 89 chairs passed...how could I have missed that....well, that's why we have peer review init...

Apart from being bad spellings of two defunct European currencies:-

So maybe there's more to this than meets the eye?

I'm thinking it's probably this guy...

https://en.wikipedia.org/wiki/Marc_Frank_Montoya

Below is a map of chair numbers, which may help. Numbers 55 and 145 are going around the end pulleys. The orientation is symmetrical, whether left/right, up/down, or increasing/decreasing chair numbers.

You could pass all other chairs but your own, but if the emkark/debark points are a bit distant from the ends of the loop, you might miss a chair or two.

...145
146-144
147-143
|
|
178-112
179-111
180-110
001-109
002-108
|
010-100
011-099
|
|
053-057
054-056
...055

This is what I mentioned earlier. You don't pass the chair behind you and you don't pass the chair in front of you and of course you don't pass the chair you are sitting in so 177 chairs are passed.

Why do you think you wouldn't pass the chair in front of you?

Passing the car in front at the getting on point would be similar.

Look at the diagram, Unlike the picture there is a chair at the end of the system when you get off and on. According to the original post, chairs 10 and 100 are right beside each other. Ten to 100 is 91 chairs. That is 45 on the down side, 45 on the upside and one on the end.

Frank is in chair 10, I posit that chair 1 is ahead of Frank, chair 11 is trailing Frank, and chair 100 is at the top opposite Frank. Therefore, the answer is: Frank passes one chair. Clue to my thinking: You might think my answer is off, but if you think about it, you will see my answer is spot on.