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### #31 capecchi

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Posted 18 September 2011 - 09:53 PM

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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### #32 thoughtfulfellow

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Posted 19 September 2011 - 01:33 AM

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.

read my false assumption
Spoiler for and note

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### #33 thoughtfulfellow

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Posted 19 September 2011 - 04:50 AM

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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### #34 capecchi

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Posted 19 September 2011 - 06:44 AM

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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### #35 thoughtfulfellow

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Posted 19 September 2011 - 07:59 AM

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.
Spoiler for answer is

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### #36 DeeGee

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Posted 19 September 2011 - 11:03 AM

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, 19 September 2011 - 11:07 AM.

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### #37 thoughtfulfellow

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Posted 19 September 2011 - 02:19 PM

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.
Spoiler for checking your math

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### #38 DeeGee

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Posted 19 September 2011 - 02:27 PM

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

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Posted 19 September 2011 - 04:26 PM

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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### #40 wolfgang

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Posted 19 September 2011 - 08:10 PM

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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