3
$\begingroup$

As shown in this question, it is trivial to schedule trains running either north-south or east-west in a square city along randomly placed (vertical and horizontal) tracks and ensure that two trains always meet at track intersections--so long as all trains have the same speed.

What are the conditions on the number and placement of vertical and horizontal tracks and train speeds such that such intersections remain even though the trains have different speeds? Must all the train speeds be related by rational numbers?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ As you say, in that other question, we are dealing with a square city. Even if you keep all speeds the same, what happens in a nonsquare rectangular city? The construction at the other question doesn't apply. $\endgroup$
    – Gerry Myerson
    Commented Oct 25, 2017 at 21:53
  • 1
    $\begingroup$ "randomly placed": you mean "arbitrary placed". There's no randomness here. Apart of this, is it nontrivial to formulate the question precisely? You don't even say that you want one train per track, you don't say that you want then to have some constant speed, you don't say whether this speed is prescribed beforehand or if you can choose it with the only requirement that all speeds are distinct; you don't say whether 2 trains should meet: once? periodically? each time they cross the other's track? $\endgroup$
    – YCor
    Commented Feb 23, 2018 at 12:44
  • $\begingroup$ One could easily use a random function to place the horizontal and vertical locations of the tracks. The other conditions (vertical/horizontal, etc.) are inherited from the cited source problem. $\endgroup$
    – David G. Stork
    Commented Feb 23, 2018 at 15:39
  • $\begingroup$ Instead of you, David, doing some extra work, you want all your readers to do it. This is, here and in general on MO, why so few people bother. $\endgroup$
    – Wlod AA
    Commented Jul 23, 2018 at 5:57

1 Answer 1

Reset to default
2
$\begingroup$

Likely one can represent this system by a set of linear equations. However, the speeds are a function of distance over time, and do not have to be related by rational numbers but the times do have to be related.

Here is a geometric model which is a little different from what you present, but is easy both to visualize and analyze, and you can adapt it to your situation. Imagine a train running on a (topologically) circular track at constant speed in 3d-space. Now pick two points on this track, and add a second circular track and train that "comes sufficiently close" to the first train at the appropriate times. Both of them have to cover the same fraction of track in the same time, but their speeds can be any ratio you desire , especially if you bend the first track. You can now pick two other points on this combination, and under mild restrictions add a third train and track so that there are meetings between each pair of trains. It is an interesting exercise to determine how to place a new circular track and a train to satisfy intersection conditions. Perhaps using a Minkowski space-time model will help visualize the conditions needed for intersections.

There was a linear version of this at Alexander Bolgomony's website cut-the-knot, but involving constant velocities, not just constant speeds. You might find it useful to consider the Four Travelers problem there.

Gerhard "Is Going Back In Time" Paseman, 2017.09.25.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Interesting approach, but I don't quite see how it solves my general problem. In yours you can adjust both the radii of the circles and speeds, guaranteeing that train 1 completes loop $1$ in time $T$ and train $2$ completes loop $2$ in time $T$. But that is fundamentally different from the problem as posed, in which the loops must be the same size. Right? $\endgroup$
    – David G. Stork
    Commented Sep 25, 2017 at 18:40
  • $\begingroup$ In the linked question, it is clear that only horizontal and vertical tracks are considered. In your question, you use the word "randomly", and it is no longer clear what other restrictions are placed on the tracks. I still think you can use the model above to think about your situation. If you have the (unclear to me, so additional) restriction that the tracks are all the same length, then I imagine you will see rational speed ratios implied. Gerhard "May Be On Different Track" Paseman, 2017.09.25. $\endgroup$
    – Gerhard Paseman
    Commented Sep 25, 2017 at 19:15
  • $\begingroup$ The tracks are north-south and east-west so certainly "randomly" refers to their separations. Nevertheless, I've clarified the question. $\endgroup$
    – David G. Stork
    Commented Sep 25, 2017 at 20:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .