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#11 wolfgang

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Posted 14 September 2011 - 02:07 PM

Spoiler for My Solution

The time needed between each two stations is one minute,except at ends where the train should do a one minute curve and get back to that station(on the other direction).
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#12 wolfgang

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Posted 14 September 2011 - 02:15 PM

I`ll give an example:
At 10:00 a.m.........a train arrives the station( from the left side).
at 10:01 a.m.........a train arrives the station( from the right side).
at 1005 a.m..........a train arrives the station( from the lsft side).
at 10:06 a.m..........a train arrives the station(from the right side).
and so on.........
All the staions are equally spaced,all the trains have the same speed,and the time needed between any two stations is one minute.
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#13 AntonMagus

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Posted 14 September 2011 - 03:03 PM

Each station has a north-bound and south-bound line and station 24 north-bound travels around a loop before pulling into station 24 south-bound. After leaving station 1 south-bound it again travels around a loop at that end before pulling into station 1 north-bound. So, with 24 stations there are 48 stops. At station 12, north-bound a train (#1) arrives, waits 1 minute and departs. Exactly 4 minutes after train #1 arrived at station 12, train #2 arrives, and 4 minutes later, train #3 and so on. The trains thus have to travel between stations for a period of exactly 3 minutes, wait 1 minute and then depart to travel another 3 minutes to the next station. So travel plus stopped time is 4 minutes. Except when they go round the loop at each end, which takes each train 5 minutes from arrival at the station to arrival back at the same station on the opposite line. So, train #1 arrives at station 24 south-bound 1 minute after train #2 north-bound arrives at station 24.
The minimum number of trains to achieve this is thus 1 for each stop, or 48 trains, but for each stopped train there could be one or more trains in transit (with all trains traveling at fractional speed), so the answer could also be any multiple of 48.

Edited by AntonMagus, 14 September 2011 - 03:08 PM.

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#14 AntonMagus

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Posted 14 September 2011 - 03:11 PM

Spoiler for My thoughts...

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#15 capecchi

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Posted 15 September 2011 - 05:24 AM

64 trains. In order for this problem to remain an interesting logic puzzle I feel we need to assume things about transit time, ie that they are all equal. Without this assumption any amount of doctoring can yield a result, and many results besides, as we can see already in these posts. With this assumption lets call the transit time between stations 'x'. Lets consider the moment when the leftward moving train just arrives at station 12, and call this train L. This means that the rightward moving train that arrived a minute earlier is just leaving headed towards station 24, lets call this train R.
Consider first train L. What we know is eventually train L will make it around the track to where train R is, and since trains arrive heading right every 4 minutes, there is an integer number of 4 minute intervals between trains R and L (heading through the end with station 1). With the convention that x is the transit time between stations (including the end 'turnaround' that takes a train from station 1 back to station 1 or 24 back to 24), then the time interval between L which just arrived at station 12 and R which is just leaving station 12 is:
23x+24
The 24 is from the fact that there are 24 stations that each require a stay of a minute (including station 12 twice since train L must wait a minute before it leaves as well as wait a minute when it returns before reaching the position where train R was. Since this must be a multiple of 4, we set:
23x+24=4n
A similar argument follows that there must be an integer number of 4 minute intervals between train R and train L as you go around station 24. This gives:
25x+24=4m
Combine these two equations to eliminate x and require that m and n be integers and the smallest pair of integers that offers a solution is n=31 and m=33. This means that there are 31 4 minute intervals from L to R and 33 4 minute intervals between R and L, or 64 total intervals of 4 minutes. 64 trains.
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#16 capecchi

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Posted 15 September 2011 - 06:06 AM

Correction to my last post. Due to an algebra error (24/4 is not 8) the equation to find solutions to is 23m=25n-12 for some integers m,n. The lowest pair satisfying this is m=n=6 which means that x=0. If we wish for a non-zero transit time the next pair of integers is n=29, m=31 giving x=4. Therefore there will be 60 trains.
Also I'm confused by the example mentioned above. It implies there is actually a 5 minute gap between trains traveling in one direction. If this is the case then most of what I mentioned in the above post holds except we must solve the equations:
23x+24=5n and
25x+24=5m
When we eliminate x we find the equation
115m=125n-48
which must be solved again, for some integers m,n. However, no such solution exists. (Easy enough to see when you consider that for any integers m,n 115m and 125n will have their last digit either 5 or 0, but when we subtract 48 from 125n we get the last digit is 2 or 7)
So if trains to have 4 minute spacing, the answer (lowest number of trains with non-zero transit time) would be 60 and I suggest a new example of:
10am... train arrives from the left
10:01... train arrives from the right
10:04... train arrives from the left
10:05... train arrives from the right
...
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#17 Phaze228

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Posted 15 September 2011 - 06:26 AM

Interestingly it's presented like a large clock. Each station or number is placed in equal distance from each other.
The one aspect that's added however is what I think....
Spoiler for -

(Though that does beg the question, how they avoid crashing...but this is a logic puzzle not a reality puzzle)
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#18 swapnil

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Posted 15 September 2011 - 08:07 AM

The time needed between each two stations is one minute,except at ends where the train should do a one minute curve and get back to that station(on the other direction).

Spoiler for Is it right?

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#19 thoughtfulfellow

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Posted 15 September 2011 - 01:58 PM

I was sitting in a subway station No. 12(toatal stations are 24).
On this line are several trains transporting passengers from staion(No. 1 ) till the end station( No. 24),then each train will continue the journey from side to side(both sides are connected together at each end).I noticed that after each 4 Minutes a train arrives this station from left side toward the right side,one minute later another train arrives the station but to the opposit direction,and so on.
If each train stays in each station exactly one minute( notice that the train at each end will make a curve and return back to that staion,just like a circle).
Howmany trains are there on this line?

The time needed between each two stations is one minute,except at ends where the train should do a one minute curve and get back to that station(on the other direction).

I must have interpreted something wrong.
Spoiler for The way I read the problem

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#20 wolfgang

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Posted 15 September 2011 - 05:58 PM

I must have interpreted something wrong.

Spoiler for The way I read the problem

the other train arrives exactly next to one which left.
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