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There's a boy. Let's call him Konrad. He's got his grandparents on father's side east of the city and on mother's side west of the city. Every Sunday, he walks to the train station and takes the first arriving carrier that happens to head towards any of the grandparents' places.

After a while, the mother's parents complain that they hardly see him, while father's parents remark that he's showing up a bit too often. Konrad verifies their story and realizes that he's been visiting the latter almost every weekend, while hardly ever visiting the former.

How is it (statistically) possible?

Some premises to clarify the circumstances for a rigid deduction process.

  1. It's not a trick question - the answer is statistically deducible.
  2. There's only one train line and each end point is at one of the grandparents'.
  3. The train in each direction arrives/leaves exactly every 60 minutes.
  4. There's never any extraordinary circumstances affecting the schedule.
  5. The track is single-lane so the trains never arrive at the same time.
  6. The boy arrives at the station at random occasions.
  7. The boy has no preferences towards any of the destinations.
  8. The grandparents are always home.
  9. There might be multiple trains in service and all travel at a constant speed.
  10. The boy can board at any non-endpoint station closest to him home.
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    $\begingroup$ Not very nice of you to ignore one set of grandparents Konrad! $\endgroup$
    – CodeNewbie
    Commented Sep 18, 2015 at 5:03
  • $\begingroup$ @CodeNewbie Well, it's all statistics, not intentional choice, so I (or the Konrad of the story) isn't at blame, hehe. $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 10:43
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    $\begingroup$ the track is single lane but there are multiple trains in service? $\endgroup$
    – Alex
    Commented Sep 18, 2015 at 14:17
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    $\begingroup$ @Alex I assume it means single-platform. There could still be passing/switching spots between stations. $\endgroup$
    – Set Big O
    Commented Sep 18, 2015 at 16:32

3 Answers 3

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The train to his paternal grandparents is closely followed by the train to the maternal grandparents. In order to go to the maternal grandparents he has to arrive in the interval between the two trains. Let's say it is 5 minutes. He will reach his maternal grandparents only if he arrives at the station during those 5 minutes. If he arrives at the station at any of the other 55 minutes. he ends up going to his paternal grandparents.

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Ok, with some of the additional clarifications, I've revised my answer, which I feel provides a tad more of an explanation about the why of the other perfectly acceptable response. This revision would apply even for a multi-track scenario, and applies no matter where Konrad's station is located relative to the endpoints - as long as there is a fixed schedule (with no simultaneous arrivals), as stated above in the criteria above.

Let X = the time between the arrivals of any west-bound train and the next east-bound train.
Let Y = the time between the arrivals of any east-bound train and the next west-bound train.

Since there is a fixed 60-minute interval for each direction, the probability of taking an east-bound train (to the paternal grandparents) should be directly related to the ratio X / 60 (with Y / 60 representing the probability of heading west). The higher X is, the lower Y is, and thus...whichever interval has the longer wait, will determine the most probable direction he travels.

Thus, for Konrad to visit the paternal grandparents more often, the schedule at Konrad's station has a larger interval between the arrival of the west-bound train and the next east-bound train, than between the east-bound and next west-bound train.

Original below (based on a very simple model of a single train which traverses the line back and forth - still works within those constraints, as it provides a practical explanation regarding the scheduling...but isn't as flexible):

Konrad lives closer to the east side of town. This is dependent upon the following:
- There is a single train (single track) which travels to each end-point, and then travels back on the same track towards the other end-point.
- The full round-trip takes 60 minutes (each direction arrives/leaves exactly every 60 minutes).

The probability of Konrad catching the train headed in any particular direction, is directly related to his proximity to either end-point.

Think of the total round trip as being split into two parts: the path from Konrad's stop to the east endpoint and back Pe, and the path from Konrad's stop to the west endpoint and back Pw. The closer he lives to the east endpoint of the train line, the shorter Pe is, and the longer Pw is. Therefore, when arriving at a random time, it's much more probable that he arrives while the train is traveling along the Pw path, in which case the train will be headed east when it finally arrives at his stop.

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  • $\begingroup$ It doesn't depend I the proximity to the endpoints generally. Under the assumption that there's only one train, it happens to coincide with the actual reason, though. However, in Konrad's city, there might be multiple trains. Or not. Or the trains could be produced at each of the endpoints and then dismantled at the other. Weird scenario but still possible. $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 6:09
  • $\begingroup$ I'd be intrigued in practical means of having multiple trains sharing a single track. You'd have to introduce multiple passing / parallel tracks, pull-off-points or similar situations. In absence of that information, I was offering a simple, straight-forward, high-probability scenario favoring the paternal grandparents. I guess if we introduce variable speed for different portions (or directions) of the track, then it would technically not align with distance, but with the time taken for each of the Pe and Pw paths, boiling down to living closer to the shorter time-period interval. $\endgroup$
    – Crumbs
    Commented Sep 18, 2015 at 7:40
  • $\begingroup$ Single track solutions are quite common in many places of the world. Where I live, we have this. At some portions, there are dual tracks, of course, but most of the distance works out surprisingly well on a single track, though. Even if we assume a constant speed, it isn't dependent of the proximity to any of the endpoints. I updated the list of premises (9 and 10). You've just happened to pick a special case (single train only) and then, the proximity coincides with the actual reason. $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 10:48
  • $\begingroup$ Thanks for that information. I've revised my answer accordingly. I realize that my answer now closely aligns with the other fine response, though I wanted to touch upon a bit more of the explanation. I was seeing the interval as being a result of the location, rather than abstracting it out to a broader scenario, where focusing on the intervals themselves allows much more flexibility. $\endgroup$
    – Crumbs
    Commented Sep 18, 2015 at 11:45
  • $\begingroup$ I suggest that you keep both the answers, because they together illuminate the trickiness of the question. +1 for for finding the special case. $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 13:28
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This is because the boy lives

in the west of the city.

And there is only one track

so there is only one train going back and forth.

The track looks like this:

W -- NW -- N -- NE -- E
GM B GF

And the schedule looks like this:

14:00 leave at station west (near mother's parents: GM)
14:05 leave at station north-west (where the boy lives: B)
14:15 leave at station north
14:25 leave at station north-east
14:30 leave at station east (grandparents on father's side: GF)
14:25 leave at station north-east
14:45 leave at station north
14:55 leave at station north-west (B again).
(repeating every hour).

So we can conclude

the boy goes to his father's parents 50/60 times, and his mother's parents 10/60 times. As he only goes to his mother's parents if he arrives between xx:55 and xx:05 at the station.

In order to go to his mother's parents more often

he has to walk to station north, further away from mother's parents!

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  • $\begingroup$ Sorry, but no. The boy can live anywhere along the line (except for the endpoints, maybe). There might be multiple trains serving the line. Kindly, view the two last items on the premise list (addition made based on another users comments). $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 13:32
  • $\begingroup$ That is not how a train schedule works. When you take a different station, trains arrive and leave at different times. Anyway, does that mean sriram answer is correct, or are we still missing a important detail in the explanation here? $\endgroup$
    – Dorus
    Commented Sep 18, 2015 at 13:38
  • $\begingroup$ Different time, of course. But the boy can live very much to the east and still end up going to father's parents most of the cases. The answer by sriram is correct, yes. $\endgroup$
    – Konrad Viltersten
    Commented Sep 18, 2015 at 13:44
  • $\begingroup$ You are talking about endpoint stations, so i assume the track is a long line, and not a circle. You also say the track is single-lane and the trains never arrive at the same time (Normally trains arrive especially at the same time on sing-lane tracks, as the station is usually the only dual-lane place where they can pass each other). That all lead me to the conclusion there is in fact, only one train. Having only one train it's almost impossible to get the described arrival times further to the east. Unless your train wait 50 minutes before departing at E. $\endgroup$
    – Dorus
    Commented Sep 18, 2015 at 15:02
  • $\begingroup$ I meant waiting 50 minutes before departing at W, making that the longest part of the route. $\endgroup$
    – Dorus
    Commented Sep 18, 2015 at 15:19

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