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

*gives a total for the time between R and L to be:*

**all transits including the ends have the same transit time**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***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?*

**If the gap is 5 or 7 minutes there is no solution**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?