[wolfgang]

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?

[bonanova]

Since we are not told anything about train speed or station distances, there could be any number of trains.

For example you could do this with a single train.

Assume the train travels at a constant speed.

Let the spacing from station 1 to station 12 be three times the spacing from station 12 to station 24.

Then it takes 1/3 the time to go from 12 to 24 and return to 12 that it takes for the train to go from 12 to 1 and back to 12.

Finally, the speed of the train would be such that a complete round trip takes 4 minutes.

You'd see the left to right train reappear from the right a minute later, and reappear from the left 3 minutes later.

Edit:

Ah, but there is a 1-minute wait at each station. The above is no longer a solution, and I have a puzzle on my hands.

[Guest]

23x4=92 +23 = 115

25x4=100 +25 = 125

48 trains

[Guest]

At first glance, one is tempted to go with the default answer of 48. This works if the stations are equally spaced where the train takes 3 minutes to travel between stations and the loop around to other directions at end requires 4 minutes. BUT: a train travel time of 1 minute between stations, keeping the 4 minute loop at end would meet specifications and yield only 26 trains on the line. Since we can't even know if the stations are equally spaced or how many trains might be on the end loop at one time, we can only arrive at a minimum number of trains.

[Guest]

There can be two cases: In first case, I assume that after station number 24, train directly comes to station 1 through some back route so that every train is moving in a circle. In this case, we can only find minimum number of trains required. There are 24 stations so every train is stopping for at least 24 minutes. Now even if train is travelling at light's speed and takes no time in covering distance between one station to other, a train will repeat itself at a particular station on only 24th minute. So there must be 6 trains every side, making a total of 12 trains. This is the minimum number. Maximum can be infinity.

However, if we assume that after station no 24, train comes back to the staion no 23 and then moves towards station no 1, then total number of stop in a round trip (station 12 to again arriving at 12 from same direction) will be 46 minutes which means, train doesn't travel at light's speed (46 not being multiple of 4), so it will take at least 2 min to cover this distance (if the train is fastest). So in that case also we will end up at having 12 trains.

Finally: Minimum number is 12. Actual number depends on speed and distance, Maximum number is infinity.

[Guest]

Minimum loop at end would be 4 minutes. With a time of 15 seconds between stations would yield 15 trains in circuit. (Yields total time for 1 train to make round trip of 60 minutes. With trains every 4 minutes means 15 trains required). For each 5 second we add to the average time between stations, we will require 1 additional train. Of course we have made the assumption that the observed 4 minutes from the arrival of one train from the left until the next trains arrival from the left is consistent for all trains,

[bhramarraj]

46 trains.

After one train passes, another train comes in after 3 minutes of its departure, (since the train stayed for 1 minute at the station). Suppose there is no other train on the track, then train leaving the station is to arrive in 3 minutes after leaving the same station which is possible only when there are two stations. Similarly if there are three stations four trains are needed to maintain the timings, and so on......! Therefore, for 24 stations, 24 outgoing and 22 incoming trains are required.

MORE interestingly, if the trains take no time from any station to other station, then the no of trains required will be only SIX......! ! !

[wolfgang]

Since we are not told anything about train speed or station distances, there could be any number of trains.

For example you could do this with a single train.

Assume the train travels at a constant speed.

Let the spacing from station 1 to station 12 be three times the spacing from station 12 to station 24.

Then it takes 1/3 the time to go from 12 to 24 and return to 12 that it takes for the train to go from 12 to 1 and back to 12.

Finally, the speed of the train would be such that a complete round trip takes 4 minutes.

You'd see the left to right train reappear from the right a minute later, and reappear from the left 3 minutes later.

Edit:

Ah, but there is a 1-minute wait at each station. The above is no longer a solution, and I have a puzzle on my hands.

Thanks...I am very proud when I see you joining my (stupid) puzzles.

[wolfgang]

At first glance, one is tempted to go with the default answer of 48. This works if the stations are equally spaced where the train takes 3 minutes to travel between stations and the loop around to other directions at end requires 4 minutes. BUT: a train travel time of 1 minute between stations, keeping the 4 minute loop at end would meet specifications and yield only 26 trains on the line. Since we can't even know if the stations are equally spaced or how many trains might be on the end loop at one time, we can only arrive at a minimum number of trains.

The stations are equally spaced, all trains have the same speed,and the distance between any two trains is also equal.

[Guest]

8

Edited September 14, 2011 by Suwandy_wu

[wolfgang]

There can be two cases: In first case, I assume that after station number 24, train directly comes to station 1 through some back route so that every train is moving in a circle. In this case, we can only find minimum number of trains required. There are 24 stations so every train is stopping for at least 24 minutes. Now even if train is travelling at light's speed and takes no time in covering distance between one station to other, a train will repeat itself at a particular station on only 24th minute. So there must be 6 trains every side, making a total of 12 trains. This is the minimum number. Maximum can be infinity.

However, if we assume that after station no 24, train comes back to the staion no 23 and then moves towards station no 1, then total number of stop in a round trip (station 12 to again arriving at 12 from same direction) will be 46 minutes which means, train doesn't travel at light's speed (46 not being multiple of 4), so it will take at least 2 min to cover this distance (if the train is fastest). So in that case also we will end up at having 12 trains.

Finally: Minimum number is 12. Actual number depends on speed and distance, Maximum number is infinity.

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).

[wolfgang]

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.

[Guest]

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 September 14, 2011 by AntonMagus

[Guest]

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.

[Guest]

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.

[Guest]

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

...

[Guest]

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....

General Idea

2 Trains at each station which is elaborated by the wording of the puzzle.

So 48 trains.

However, there's a one minute gap in between the opposing directions of the traveling trains. Which means that only 24 trains are traveling in the same direction on the line at once. However, that is only for a minute, the next 3 minutes, 48 trains are traveling on the line, due to the one minute gap.

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

[Guest]

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).

Now it's much better. We need to calculate total time spent between a train comes to station number 12 and comes back to the same station from same direction for the second time. It will stop at 48 stations, twice at each station (even at both ends). So time taken = 48 minutes. Now, it takes one minute between every two stations. So time spent = 46 minutes + 2 minutes at ends = 48 minutes. So total time = 96 minutes. Hence total number of trains = 96/4 = 24 trains. I hope this is fine.

But my problem didn't end. Let's say at 00:00, a train arrives at station number 12 from left side. So at the same moment, a train should be arriving at station number 10 also from left side itself (so that it will stop there for a minute; will reach 11 in a minute; stop there for a minute; and will reach 12 at 4th minute). Similarly trains will be arriving at stations from left side stations numbers 2,4,6,8,10,12,14,16,18,20,22,24. Now to arrive a train at station number 2 from left side again at 4th minute a train must be arriving at station number 1 from right side (so that it will stop there for a minute; will make circle for a minute, will stop again for a minute and then will go to station number 2 in 4th minute). Following a similar way, we can say that in that case, Trains should be arriving at stations numbers 1,3,5,7,9,11,13,15,17,19,21,23 from right direction at 00:00. Now this creates a problem. There is no way that a train can reach station number 12 from right direction at 00:01 (as in that case, train should be departing from station number 13 at 00:00, not arriving there). I am somewhere short of 2 minutes.

[Guest]

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.

1 minute after a train arrives from the left it leaves the station at the same instant that a different train arrives from the right. This timing should also be true for station 24. Since the time between trains is 4 minutes, this train leaving the station must arrive from the right in exactly 4 minutes, making the loop 4 minutes regardless of other restrictions. If the end loop is 1 minute as the latter post indicates, then are you saying that the train from the right arrives 1 minute after the train from the left leaves the station, which would be 2 minutes after the arrival of the train from the left?

[wolfgang]

I must have interpreted something wrong.

1 minute after a train arrives from the left it leaves the station at the same instant that a different train arrives from the right. This timing should also be true for station 24. Since the time between trains is 4 minutes, this train leaving the station must arrive from the right in exactly 4 minutes, making the loop 4 minutes regardless of other restrictions. If the end loop is 1 minute as the latter post indicates, then are you saying that the train from the right arrives 1 minute after the train from the left leaves the station, which would be 2 minutes after the arrival of the train from the left?

the other train arrives exactly next to one which left.

[wolfgang]

I`ll give another example, in more details:

....................................................Arrival.............................................Departure..............................

Left side........................................6:00...................................................6:01..................................

Right side......................................6:01...................................................6:02..................................

6:03.................No train............

6:04.................No train............

6:05.................No train............

6:06.................No train............

Left side.......................................6:07....................................................6:08..................................

right side......................................6:08....................................................6:09..................................

and so on................

[bhramarraj]

In the original question it looks as if one train comes on the station at say 6:00 from left side, then another train comes from left side at 6:4.

[Guest]

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?

I`ll give another example, in more details:

....................................................Arrival.............................................Departure..............................

Left side........................................6:00...................................................6:01..................................

Right side......................................6:01...................................................6:02..................................

6:03.................No train............

6:04.................No train............

6:05.................No train............

6:06.................No train............

Left side.......................................6:07....................................................6:08..................................

right side......................................6:08....................................................6:09..................................

and so on................

Hence, I agree with bhramarraj that this would mean, in your example above, that the next train from the left would arrive at 6:04. Did we loose something in translation?

[Guest]

I agree there's something that needs to be cleared up. In the original post it sounds as if trains are arriving FROM THE LEFT every 4 minutes. The first example posted by wolfgang then makes it seem that trains arrive FROM THE LEFT every 5 minutes. Now this most recent example shows they arrive FROM THE LEFT every 7 minutes.

So the question that needs to be clarified: How often do trains arrive FROM THE LEFT?

[Guest]

I`ll give another example, in more details:

....................................................Arrival.............................................Departure..............................

Left side........................................6:00...................................................6:01..................................

Right side......................................6:01...................................................6:02..................................

6:03.................No train............

6:04.................No train............

6:05.................No train............

6:06.................No train............

Left side.......................................6:07....................................................6:08..................................

right side......................................6:08....................................................6:09..................................

and so on................

To be curt...

is it

Train to station = 4 minutes

Stationed train = 1minute

Opposite train arrival = the time of the departing train

Train curving from station 24 to the original station (No.1 infer) = 5 minutes?

Edited September 16, 2011 by Phaze228

[Guest]

i know its a trick question HAHAHAHAHAHA! good one