Apartment and The Lifts 2

Algebra Level 5

There is an apartment with 14 floors, the floors are numbered from 0 to 13 . You’re in this apartment, your floor is unknown but you aren’t on Floor 0. There are 2 lifts and you have to get to Floor 0 urgently. The lift you choose will first come to your floor (or if it’s already in that floor it doesn’t need to) and than get to 0. How can you make the maximum distance as short as they can be ? Which floors should the lifts be at? (Lifts can be in separate floors.)

Write the answer like first the smaller floor, then the upper floor. For example:

If the lifts are on the floors 2 and 5, write 25

Or if the lifts are both on Floor 10, type 1010

Love it? Then, here is the 1st problem of “Apartment and The Lifts”!

The answer is 1213.

1 solution

Ege Şik
Apr 11, 2021

13. Floor (LIFT)	0 to here, 13 to 0	13
12. Floor (LIFT)	0 to here, 12 to 0	12
11. Floor	1 to here, 11 to 0	12
10. Floor	2 to here, 10 to 0	12
9. Floor	3 to here, 9 to 0	12
8. Floor	4 to here, 8 to 0	12
7. Floor	5 to here, 7 to 0	12
6. Floor	6 to here, 6 to 0	12
5. Floor	7 to here, 5 to 0	12
4. Floor	8 to here, 4 to 0	12
3. Floor	9 to here, 3 to 0	12
2. Floor	10 to here, 2 to 0	12
1. Floor	11 to here, 1 to 0	12
0. Floor

The most risky one is the floor 13 , so we should first find a solution for it. Because we should guarantee the quickness, and we always have to get down 13 floors, if there isn’t a lift on 13, it will get even worse! So one of the lifts should be on 13th floor. What about the second one? We will try to make duration even faster, but the longer duration left (12th Floor) should be the fastest so we should start from 12. The other lift to 12th floor.

In conclusion, one of them should be on 12th floor and the other one on the 13th floor. This way, if you are on 13th floor, you have to pass 13 floors and if you’re on another floor, you should pass 12 floors.

But the statement isn't correct! You said it should be the fastest for the longest route. This is sufficed by any pair with $13$ so there are $13$ correct answers! And if we want to be as quick as possible on average, then the correct combination is either $5-13$ or $4-13$ , which both result in average duration of $9$ units of time. And this is the fastest it can happen.

Veselin Dimov - 2 months ago

you should report if the answer is wrong.

Aditya Mittal - 2 months ago

I'm not sure if my reasoning is correct. I'd like to see more people who agree with me.

Veselin Dimov - 2 months ago

Hi! We both know that 13th floor to 0 means at least 13 units, right? So I agree with you that one of the lifts should be at 13th floor. You’d be right if I said average, but I meaned the “maximum distances”, they should be the minimum. So if the Lift 2 is on 4th floor or 5th floor, it means if you are on floors 12, 11, 10 or 9, it will last for 13 unites of time. But if Lift 2 is on 12th floor, they will last 12 unites of time.

Ege ŞIK - 1 month, 2 weeks ago

this is totally illogical. If the person is on the 13th floor the best place for the lift is at 13, obviously. but what if the person is on the 1st floor? the elevators are at 12 and 13, He will have to wait 6 or 7 units more than necessary. the correct answer is 13 and 6. floors 10-7 would be served by the lift at floor 6, 11-13 served by the lift at fl 13. this way, 13,13:12,13,11:13, 10:13,9:12, 8,10,7:8,6:6,5:6,4:6,3:6,2:6,1:6.

Jonathan Gordon - 1 month, 2 weeks ago

Please pay attention to the words in bold in the problem: “maximum distance”. Yes, the 6th-floor-lift shortens the unnecessary distances for the lower floors, but what if you are on the 12th floor? Or 11th floor or 10th floor? They will be even worse, 13 units of time and you don’t want to get risk. But if the second lift is on 12th floor, all of the floors (except 13th floor) will last 12 units of time. Even if it causes unnecessarily long distances for lower floors, the worst floor ’s distance is as short as it can, and this was our purpose. To make the worst floor better as much as we can.

Ege ŞIK - 1 month, 2 weeks ago

The correct answer can only be if the lifts are on the 3rd and 10th floor. That is the most optimized case. You can check the proof here:

https://docs.google.com/document/d/1vTr5TYBmPKsP_uAYVPjR4OykPy1R1D9uFACGiBekHq8/edit?usp=sharing

Harshit Awasthi - 1 month, 1 week ago

This way if you are on 13th floor, it will last at least 16 units of time. Also on 12th floor 14 units of time.

Ege ŞIK - 1 month, 1 week ago

Your statement would be true if I said minimum distance to you but I said minimum time for “first taking you and then bringing you to 0”

Ege ŞIK - 1 month, 1 week ago

Maybe you should change the statement to "maximum distances as short as they can be".

Veselin Dimov - 1 month ago

@Veselin Dimov – Is it okay now?

Ege ŞIK - 1 month ago

If we assume the Lift travels with uniform speed or it's the same for each and every floor it needs to cross(Either Upwards or Downwards). If the above is true then only the minimum number of floors will produce minimum time.

Harshit Awasthi - 1 month ago

Ahhh know that my English phrases isn’t as good as you so I don’t understand if you agree with me but it seems like you don’t. I want to say that, before you post any “this-statement-is-wrong-message”, please check which floor needs how many units of time. I don’t know how to explain. Hmmm, I have an idea! Think like you have to bring someone to the hospital urgently (May God care you. Of course I don’t want something like this to happen) yes, it’s good to be there in short time, but what if you are on a worse floor? I mean, you don’t want to risk it at all. We have to think about the worse scenario.

Ege ŞIK - 1 month ago

Apartment and The Lifts 2

The answer is 1213.

1 solution

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