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## Question

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?

## 55 answers to this question

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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?

Sorry about the mistake...this example is the right one...i.e. 2 trains(left then right,in one minute interval),then 4 minutes(No trains),and so on.

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Alright then. If the new example is correct and minute by minute we go: train from left, train from right, no train, no train, no train, no train, train from left, etc. Then We need to solve:

23x+24=6n and

25x+24=6m

This gives 23m=25n-8

Which has a solution of m=n=4 but this again gives x=0 so if we want the first set of integers that give us a nonzero transit time we find:

m=29,n=27. This gives x=6 minutes. So the track has 56 trains, each 6 minutes apart. For explanation of what I did here see my first post.

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I (and, I think others) have made a false assumption:

that loop at each end is equal which is not possible with the specifications given.

with the 1 minute between trains as specified by Wolfgang in post #12, there are 25 trains. The easy to overlook error is that the loop at the station 1 end is 2 minutes while the loop at station 25 end is 4 minutes, otherwise the specification of train arriving from right at the same time as a train leaves station headed toward the right is not satisfied. Now we see to go from arriving at station 1 to leaving station 24, the train is in stations for 24 minutes and between stations 23 minutes for a total of 47 minutes. Same time on other side plus loops - 47+47+2+4=100. Divide by 4 since trains must be spaced at 4 minute intervals and answer is 25.

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I (and, I think others) have made a false assumption:

that loop at each end is equal which is not possible with the specifications given.

with the 1 minute between trains as specified by Wolfgang in post #12, there are 25 trains. The easy to overlook error is that the loop at the station 1 end is 2 minutes while the loop at station 25 end is 4 minutes, otherwise the specification of train arriving from right at the same time as a train leaves station headed toward the right is not satisfied. Now we see to go from arriving at station 1 to leaving station 24, the train is in stations for 24 minutes and between stations 23 minutes for a total of 47 minutes. Same time on other side plus loops - 47+47+2+4=100. Divide by 4 since trains must be spaced at 4 minute intervals and answer is 25.

If they are 25 trains,,,and there are only 24 stations,,,that means there should be only one minute interval between train and the next one,and with the extra train (No.25) would be no interval between it and the next train.

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I'm getting frustrated. How many minutes are there between trains? I see support in this thread for 4,5,6, and 7 minutes. Another way to phrase the question: If I'm only considering trains traveling to the right and I see a train pass by, how long do I have to wait to see the next one?

I've answered for 4,5, and 6 trains the answers being, respectively, 60, no solution, and 56.

Now lets go through it again for 7 trains and someone tell me if I'm making a logical error somewhere. We take a snapshot of the situation one minute after the rightward moving train arrives at station 12. At this moment the rightward moving train (call it R) is just departing. Also a leftward moving train (call it L) is just arriving at station 12. Between L and R there are 24 stations (12,11,10,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,11,12) requiring 1 minute each or 24 minutes. There are also 23 transits that have to be made (12-11,11-10,10-9,9-8,8-7,7-6,6-5,5-4,4-3,3-2,2-1,1-1,1-2,2-3,3-4,4-5,5-6,6-7,7-8,8-9,9-10,10-11,11-12) each requiring some unknown amount of time x. So the total time between trains L and R is:

23x+24

This time must be some multiple of 7 (if we are to assume there are 7 minutes between trains and equal times between all stations as we should) so:

23x+24=7n for some integer n

Similarly there are 24 stations (13,14,15,16,17,18,19,20,21,22,23,24,24,23,22,21,20,19,18,17,16,15,14,13) between R and L for a total of 24 minutes, plus 25 transits (12-13,13-14,14-15,15-16,16-17,17-18,18-19,19-20,20-21,21-22,22-23,23-24,24-24,24-23,23-22,22-21,21-20,20-19,19-18,18-17,17-16,16-15,15-14,14-13,13-12) of x minutes each so that all transits including the ends have the same transit time gives a total for the time between R and L to be:

25x+24

again a multiple of 7 so:

25x+24=7m for some integer m

Since x need not be a whole number we can use the two equations to eliminate x and get an equation relating m,n. After algebra:

175n-161m=48

No such integers exist.

To see for yourself, if you have access to something like Mathematica I used the code:

FindInstance[25 7 n - 23 7 m == 48, {n, m}, Integers]

Therefore I maintain if the gap is 4 minutes there are 60 trains.

If the gap is 6 minutes there are 56 trains.

If the gap is 5 or 7 minutes there is no solution which satisfies the given data (while requiring all transit times to be equal which I still maintain is the only way for this to be a meaningful thing to do). I think it's silly there's so much runaround on this forum. Anyone see an issue with my handling of this problem?

Also, to reiterate what I've mentioned in previous posts, these answers are for the fewest trains possible given a non-zero transit time.

Would like to hear comments about my work. Is it clear enough?

I'd like some backers for my answers... anyone wanna support my math?

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If they are 25 trains,,,and there are only 24 stations,,,that means there should be only one minute interval between train and the next one,and with the extra train (No.25) would be no interval between it and the next train.

That, with 2 minute loop at end at station 1 and 4 minute loop at end at station 24, the moment a train arrives at station 1, a train is leaving station 2. Son we have 12 trains leaving even numbered stations and 12 trains arriving at odd number stations.. Now examining closer end at station 24, with a train leaving since it is even. The next rain from the right will arrive in three minutes which puts another train in the 4 minute loop.

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I'm getting frustrated. How many minutes are there between trains? I see support in this thread for 4,5,6, and 7 minutes. Another way to phrase the question: If I'm only considering trains traveling to the right and I see a train pass by, how long do I have to wait to see the next one?

I've answered for 4,5, and 6 trains the answers being, respectively, 60, no solution, and 56.

Now lets go through it again for 7 trains and someone tell me if I'm making a logical error somewhere. We take a snapshot of the situation one minute after the rightward moving train arrives at station 12. At this moment the rightward moving train (call it R) is just departing. Also a leftward moving train (call it L) is just arriving at station 12. Between L and R there are 24 stations (12,11,10,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,11,12) requiring 1 minute each or 24 minutes. There are also 23 transits that have to be made (12-11,11-10,10-9,9-8,8-7,7-6,6-5,5-4,4-3,3-2,2-1,1-1,1-2,2-3,3-4,4-5,5-6,6-7,7-8,8-9,9-10,10-11,11-12) each requiring some unknown amount of time x. So the total time between trains L and R is:

23x+24

This time must be some multiple of 7 (if we are to assume there are 7 minutes between trains and equal times between all stations as we should) so:

23x+24=7n for some integer n

Similarly there are 24 stations (13,14,15,16,17,18,19,20,21,22,23,24,24,23,22,21,20,19,18,17,16,15,14,13) between R and L for a total of 24 minutes, plus 25 transits (12-13,13-14,14-15,15-16,16-17,17-18,18-19,19-20,20-21,21-22,22-23,23-24,24-24,24-23,23-22,22-21,21-20,20-19,19-18,18-17,17-16,16-15,15-14,14-13,13-12) of x minutes each so that all transits including the ends have the same transit time gives a total for the time between R and L to be:

25x+24

again a multiple of 7 so:

25x+24=7m for some integer m

Since x need not be a whole number we can use the two equations to eliminate x and get an equation relating m,n. After algebra:

175n-161m=48

No such integers exist.

To see for yourself, if you have access to something like Mathematica I used the code:

FindInstance[25 7 n - 23 7 m == 48, {n, m}, Integers]

Therefore I maintain if the gap is 4 minutes there are 60 trains.

If the gap is 6 minutes there are 56 trains.

If the gap is 5 or 7 minutes there is no solution which satisfies the given data (while requiring all transit times to be equal which I still maintain is the only way for this to be a meaningful thing to do). I think it's silly there's so much runaround on this forum. Anyone see an issue with my handling of this problem?

Also, to reiterate what I've mentioned in previous posts, these answers are for the fewest trains possible given a non-zero transit time.

Would like to hear comments about my work. Is it clear enough?

I'd like some backers for my answers... anyone wanna support my math?

You are right that there has been conflicting information. Wolfgang, who started this puzzle, did state that the trains transverses the distance between stations in 1 minute. The solution that I gave in post #29 on page 3 also assumes exactly 4 minutes between the arrival of a train and the arrival of the next train and that a train arrives from the right at exactly the same time as a train is leaving to the left, this being 1 minute after it arrived from the left. For the time in the loops, I made it the minimum that I could figure possible and retain this train spacing at the ends. For additional clue to my solution, check the spoiler on post #29 on page 3 labeled "Spoiler for False assumption made"

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Yeah but that's not an interesting question. If we look again at the moment there's a train leaving station 12 headed right and at the same moment a train arriving at station 12 from the left that means from station 12 to 1 and back to 12 there are 24 minutes spent at each station, then 22 minutes spent transiting between stations plus an extra unknown quantity for the loop at the end by station 1(call it L). So 46 minutes plus L on the left end of the track. Similarly, from station 12 to 24 and back to 12 we spend 24 minutes in the station, 24 minutes in transit between stations, plus some unknown quantity at the end by station 24 (call it R). So 48 minutes plus R on the right side of the track. What are we saying these days, 7 minutes between trains? It doesn't much matter since I have complete control now by being able to choose what R and L are. So if the time between trains is 7 minutes then 48+R has to be some multiple of 7. Why not make it 49, so R=1 and there are 7 trains on the right half of the track. Then 46+L has to be a multiple of 7, 49 is the closest again, so let R be 3 minutes and we have another 7 trains on the left side of the track. 14 trains.

Unsatisfying though, since I could just pick whatever I wanted R and L to be to make it fit the model we are given.

How bout we figure this out again: try this-

There are 24 stations, we're at station 12. Every 8 minutes a train arrives from the left (for example, 10am=train arrives from the left, 10:08=train arrives from the left, etc) and TWO minutes after a train arrives from the left a train arrives from the right (so 10:02=train arrives from the right, 10:10=train arrives from the right, etc). If all transit times are equal including the time going around the loops at the end, what is the fewest number of trains on the track (excluding the case of a zero transit time between stations)? Recall, upon reaching station 1 (or station 24) a loop is made that returns you to station 1 (or 24) where you have to spend another minute at the station.

Does this clear everything up?

I have a solution to this. Anyone else? I changed it slightly so as to make everyone change their answers and think it through again.

What if trains arrive every 4 minutes from the left (so 10:00=train from left, 10:02=train from right, 10:04=train from left, 10:06=train from right, etc)? An interesting question to ask yourself before you solve it is whether you think that this will require more or fewer trains than in the 8 minute case.

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Yeah but that's not an interesting question. If we look again at the moment there's a train leaving station 12 headed right and at the same moment a train arriving at station 12 from the left that means from station 12 to 1 and back to 12 there are 24 minutes spent at each station, then 22 minutes spent transiting between stations plus an extra unknown quantity for the loop at the end by station 1(call it L). So 46 minutes plus L on the left end of the track. Similarly, from station 12 to 24 and back to 12 we spend 24 minutes in the station, 24 minutes in transit between stations, plus some unknown quantity at the end by station 24 (call it R). So 48 minutes plus R on the right side of the track. What are we saying these days, 7 minutes between trains? It doesn't much matter since I have complete control now by being able to choose what R and L are. So if the time between trains is 7 minutes then 48+R has to be some multiple of 7. Why not make it 49, so R=1 and there are 7 trains on the right half of the track. Then 46+L has to be a multiple of 7, 49 is the closest again, so let R be 3 minutes and we have another 7 trains on the left side of the track. 14 trains.

Unsatisfying though, since I could just pick whatever I wanted R and L to be to make it fit the model we are given.

How bout we figure this out again: try this-

There are 24 stations, we're at station 12. Every 8 minutes a train arrives from the left (for example, 10am=train arrives from the left, 10:08=train arrives from the left, etc) and TWO minutes after a train arrives from the left a train arrives from the right (so 10:02=train arrives from the right, 10:10=train arrives from the right, etc). If all transit times are equal including the time going around the loops at the end, what is the fewest number of trains on the track (excluding the case of a zero transit time between stations)? Recall, upon reaching station 1 (or station 24) a loop is made that returns you to station 1 (or 24) where you have to spend another minute at the station.

Does this clear everything up?

I have a solution to this. Anyone else? I changed it slightly so as to make everyone change their answers and think it through again.

What if trains arrive every 4 minutes from the left (so 10:00=train from left, 10:02=train from right, 10:04=train from left, 10:06=train from right, etc)? An interesting question to ask yourself before you solve it is whether you think that this will require more or fewer trains than in the 8 minute case.

If we have 8 minutes between trains from the left, the trains are spaced at 8 minute intervals around the entire course. By your specification of the changed problem, a train will arrive at station 24 2 minutes later. Further, this train, for a minimum loop duration, must be exactly 8 minute interval ahead of the train that arrived at station 24. and 2 minutes from station 24 headed in the other direction. So loop at right plus time in station equals 10 minutes. Subtracting time in station makes this loop 9 minutes minimum. But at station 1, when a train arrives from the left, we have 2 minutes until the arrival of a train from the right and 6 more minutes until this train returns to station 1 from the left. Subtracting 1 minute for time in station for the train coming around this left loop, we have that this loop must be a minimum of 5 minutes. Since 5 does not equal 9, your premise of all intervals equal will not work. The closest we can come is the formula adding left loop plus right loop plus 24 minutes in station plus 22 times interval between stations. Dividing this by 8 gives the number of trains. Just adjust time between stations so this is evenly divisible by 8.

If we assume intervals must be integral, 3 minutes between stations and 13 trains. If we allow less than a minute, then we could have 5 trains with 5.4545... seconds (1/11 minutes) between stations.

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I didnt see the spoiler button, so admin pls put this in a spoiler.

The time to travel between stations is 1 min and the train stops for 1 min and loop time is also 1 min.

For a given train, starting from station 1, and coming back to station 1, the total time to go "full circle" is

1 (stop at station 1) + 23*2 (reach last station #24) + 1 (turn around) + 23*2 (go back to station 1) + 1 (trun around) = 49 mins

Since a train arrives at every 5 mins, the total number of trains running is 49/5 = 10 trains

Edited by DeeGee

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I didnt see the spoiler button, so admin pls put this in a spoiler.

The time to travel between stations is 1 min and the train stops for 1 min and loop time is also 1 min.

For a given train, starting from station 1, and coming back to station 1, the total time to go "full circle" is

1 (stop at station 1) + 23*2 (reach last station #24) + 1 (turn around) + 23*2 (go back to station 1) + 1 (trun around) = 49 mins

Since a train arrives at every 5 mins, the total number of trains running is 49/5 = 10 trains

There are 2 ways to do the spoiler. First way: The S that is inside a box, located slightly right and below the smiley face. The other way (and many more formatting options) is to use Special BB Codes. Special BB Codes button is just to the left of Font.

You have 1+23*2+1+23*2+1. By my calculation, this is 95. The original problem as stated by Wolfgang was that a train arrives every 4 minutes. Your assumption that the loop can be 1 minute is also wrong. I made the same mistake when I first looked at the problem. Fact is they cannot be the same time interval. I am also curious in your solution how 9.8 trains can suddenly become 10. The correct answer must be an integer.

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You are right. I didnt add the second 23*2. It is indeed 95 mins and 19 trains then.

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You are right. I didnt add the second 23*2. It is indeed 95 mins and 19 trains then.

You are ignoring the fact that a train arrives from the other direction 1 minute after a train arrives from the left and you are still using 5 minutes between trains while the problem stated 4 minutes between trains. You must calculate what time is spent in each loop as well for they cannot be 1 minute with the 1 minute interval between arrival from left and arrival from right.

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The time at each loop is ONE minute also...

The spaces between trains are always equall...that means,,all the trains depart at the same time(as if they are connected together).

The time needed to travel between each two stations is ONE minute...

Each train stays one minute in each station.

and again....one train comes to the station...stays one minute...as it leaves the station,another train reachs it...stays one minute,and will leave it.....4 minutes after that there is no train.and then the process will be repeated again as mentioned above.

The train at station 24 will stay there one minute...then it will make a one minute loop,and returns back to this station(24),the same is true for the station number 1.

I hope it is clear now...

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The time at each loop is ONE minute also...

The spaces between trains are always equall...that means,,all the trains depart at the same time(as if they are connected together).

The time needed to travel between each two stations is ONE minute...

Each train stays one minute in each station.

and again....one train comes to the station...stays one minute...as it leaves the station,another train reachs it...stays one minute,and will leave it.....4 minutes after that there is no train.and then the process will be repeated again as mentioned above.

The train at station 24 will stay there one minute...then it will make a one minute loop,and returns back to this station(24),the same is true for the station number 1.

I hope it is clear now...

Let me again explain part of the problem with your scenario. You clearly state here that a train arrives from left at station 24. Here let me set my stop watch to 0:00. Next event train leaves this station into loop at the same moment a train arrives from the loop heading back to the left, stop watch reads 0:01. Train traverses the loop and again arrives at the station, stop watch reads 0:02. This would put the train spacing heading to the left at 1 minute. Note also that the train that arrived at 0:01 from the loop had to leave station 24 at 0:00, the moment that a train arrived from the left so that train, too, would be at 1 minute intervals. A similar problem occurs at the loop to the left of station 1 with this train spacing. Before the problem can be solved, this conflict in times must be addressed.

Summarizing: At station 24, 0:00 train 'A' arrives from left, 0:01 Train 'B' arrives from right(out of loop) & train 'A' exit station to the right into the loop, 0:02 Train 'A' arrives from the right. Position of train 'B' at 0:00 had to be leaving station 24 into the loop.

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You are right. I didnt add the second 23*2. It is indeed 95 mins and 19 trains then.

When I saw 1 problem with yours, I did not examine further but just noticed you also skipped 1 minute stay at station 24 after train transverses loop. Another way to look at it. Since there are 24 stations and train stops at each twice, this is 48 stops with 48 spaces between stops, 96 intervals. Now the problem only comes to rationalizing what the loop time must be to meet all criteria. Assuming we were not told that a train arrives from the right at the same time a train leaves the station heading right. Then at station 24, the time between the train leaving the station and arriving to head the other way is the loop time. If we made the assumption that the loop time was 1 minute, then a train arrives from the right 2 minutes after it arrives from the left but we were told that this should only be 1 minute. If we are allowed to change this time interval to 2 minutes, then indeed it is an elementary problem and the answer would be 96/4=24 trains.

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When I saw 1 problem with yours, I did not examine further but just noticed you also skipped 1 minute stay at station 24 after train transverses loop. Another way to look at it. Since there are 24 stations and train stops at each twice, this is 48 stops with 48 spaces between stops, 96 intervals. Now the problem only comes to rationalizing what the loop time must be to meet all criteria. Assuming we were not told that a train arrives from the right at the same time a train leaves the station heading right. Then at station 24, the time between the train leaving the station and arriving to head the other way is the loop time. If we made the assumption that the loop time was 1 minute, then a train arrives from the right 2 minutes after it arrives from the left but we were told that this should only be 1 minute. If we are allowed to change this time interval to 2 minutes, then indeed it is an elementary problem and the answer would be 96/4=24 trains.

As you mentioned above...there are 48 stops....and if there are 24 trains(as you think),so they should be(train........train......train.....)and that means there will be no interval between the trains arriving the station number 12..

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I think that the post does'nt make sense

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.

Considering forward and backward line...

Since at the back line train will arrive at each station every 4min.

But the same is not true for forward line.(5min difference)

Since the forward and backward paths are connected via a 1min distance(same as distance between each station)

there is no solution(illogical)[speed of train and distances being same].

So i think on the forward line the difference in arrival should have been 3 min.

This will ensure the sync.

So at a given time each station would contain a train.

For instance trains at even stations would travel in forward direction

And trains at odd station would travel in backward direction.

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Key here is each train only stops in each station for one minute. Therefore as it goes through journey it does not always stop.

Assume train leaves 12 at 10:01 after stopping for 1 minute and next arrives at 10:05 then this second train has been travelling for 4 minutes since it last stopped. A minute ago it was at 11 and a minute earlier 10. A further minute earlier at 9 where it passed through and minute earlier at 8 where it stopped

So at 10 train b arrives at 8

10:01 b leaves 8

10:02 b passes 9

10:03 b passes 10

10:04 b passes 11

10:05 b stops at 12

Thus when a is at 12, we have another at 8,4,16,20 - 5 trains. There is also one more at 1 which next stops at 4, and another at 24 about to head back after it's outward loop.

In each case on the line going the other way is train passing through except for 1 and 24.

In total this gives 12 trains.

Accordingly a train follows this order 4,8,12,16,20,24,21,17,13,9,5,1,3,7,11,15,19,23,22,18,14,10,6,2,4

Or something like this

10:03 b passes 11

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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?

Key here is each train only stops in each station for one minute. Therefore as it goes through journey it does not always stop.

Assume train leaves 12 at 10:01 after stopping for 1 minute and next arrives at 10:05 then this second train has been travelling for 4 minutes since it last stopped. A minute ago it was at 11 and a minute earlier 10. A further minute earlier at 9 where it passed through and minute earlier at 8 where it stopped

So at 10 train b arrives at 8

10:01 b leaves 8

10:02 b passes 9

10:03 b passes 10

10:04 b passes 11

10:05 b stops at 12

Thus when a is at 12, we have another at 8,4,16,20 - 5 trains. There is also one more at 1 which next stops at 4, and another at 24 about to head back after it's outward loop.

In each case on the line going the other way is train passing through except for 1 and 24.

In total this gives 12 trains.

Accordingly a train follows this order 4,8,12,16,20,24,21,17,13,9,5,1,3,7,11,15,19,23,22,18,14,10,6,2,4

Or something like this

10:03 b passes 11

I do like your theory; however, it goes against the original stated problem which I have quoted above. Note the line that reads:

"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)"

Note that he says "each train" but by your method, some trains don't stay in station one minute and would violate the "each" part.

Also, examining your solution closer, you have trains stopping at station, skipping 3 stops, then stopping except at the end turnarounds. Remember from 24, train goes back to 24 (skips), 23 (skips) 22 (skips) and 21 keeping pattern, then at the nest end 1 back to 1 (skips), 2 (skip) then stop on 3 breaking your pattern. You then break your pattern again at the low end before returning to sta 4 after skipping 4 stops. Interestingly, these error due cancel out each other. so that the corrected order wold become:

4,8,12,16,20,24,21,17,13,9,5,1,4

You have also omitted the information about second train arriving from right 1 minute after the arrival of train from left. If we go by your theory of non-stops, then your 12 trains needs to be adjusted to account for those stopping in opposite direction and some explanation as to why stations 2,3,6,7,10,11,14,15,18,19,22,23 have no trains stopping. Assuming these are not redundant stations, then train number derived must also be doubled. Note those coming in from left with same pattern would mess up pattern from left when circling back and trains that do not follow the pattern are subject to causing collisions.

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Fair point - it was too late when went for first attempt.

If each train stops once only at each station then when it stops at one it loops back but does not stop so therefore you have at each end two consecutive journeys without a stop.

Consider the train arriving at 1 from 2. It waits 1 minute and travels the loop in next minute, importantly does not stop and takes third minute to arrive at 2. It waits for a 4th minute travels for 1 minute to arrive at 3 after 5. And so until it arrives at 24 after 47. Minute 48 it sits at 24. Minute 49 it loops to 24 but does not stop and minute 50 it arrives at 23, 52 at 22 and so on until arrives back at the beginning after 94 minutes.

The only question now is what the gap is between trains. The problem is not clear so perhaps someone can clarify but based on this I would be guessing something like 19 trains.

I am late for something so hopefully someone can fill in what I have missed.

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Fair point - it was too late when went for first attempt.

If each train stops once only at each station then when it stops at one it loops back but does not stop so therefore you have at each end two consecutive journeys without a stop.

Consider the train arriving at 1 from 2. It waits 1 minute and travels the loop in next minute, importantly does not stop and takes third minute to arrive at 2. It waits for a 4th minute travels for 1 minute to arrive at 3 after 5. And so until it arrives at 24 after 47. Minute 48 it sits at 24. Minute 49 it loops to 24 but does not stop and minute 50 it arrives at 23, 52 at 22 and so on until arrives back at the beginning after 94 minutes.

The only question now is what the gap is between trains. The problem is not clear so perhaps someone can clarify but based on this I would be guessing something like 19 trains.

I am late for something so hopefully someone can fill in what I have missed.

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Okay so I assumed 94 minute round trip but stopping at stations going both directions except 1 and 24.

If they stop at every station for 1 minute only then I need to remove 22 minutes from the loop.

If we are to treat the trains at equal gaps then we need to divide 72 minutes into Segments.

Wolfgang - your time gap example skipped 6:02

which leads us to 7 minute gaps. Restating this

makes it 6 minute gaps.

Thus we have 12 gaps between trains.

Because it is loop we therefore 12 trains.

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Okay so I assumed 94 minute round trip but stopping at stations going both directions except 1 and 24.

If they stop at every station for 1 minute only then I need to remove 22 minutes from the loop.

If we are to treat the trains at equal gaps then we need to divide 72 minutes into Segments.

Wolfgang - your time gap example skipped 6:02

which leads us to 7 minute gaps. Restating this

makes it 6 minute gaps.

Thus we have 12 gaps between trains.

Because it is loop we therefore 12 trains.

The 6 min. gap is correct...but the number of trains is not

Edited by wolfgang

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Okay - because the gaps are 6 mins - confirmed I could be boring and guess all multiples of 6 but the answer must lie in why the trains from each direction are a minute out of sync.

This must be caused by the terminals 1 and 24. I have assumed incorrectly that they do not stop for a minute each side of the loop.

If in fact they do and trains stop for 1 minute each direction then I go back to my 94 minutes but add on the extra minute for each end thus arriving at 96 minutes for the circuit and 6 minute intervals. The extra minute at each end also ensures the stagger as they pass 12.

This leaves me with 16 trains. With hindsight when you stop over thinking the problem and try to write the problem in plainer English it seems more obvious, so it busts remains Wolfgang that should point out why this one is still not right!

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