How many trains?
Around the perimeter of Puzzlaria, there is a circular railway with trains running in both directions. Trains run every 15 minutes each way. At 8PM, they each board a train, DualAspect heading in a clockwise direction around the city, and Ross heading counter-clockwise. Clockwise trains take 2 hours to accomplish one lap and counter-clockwise trains require 3 hours.
Including the train each one is riding on, how many trains did DualAspect and Ross see on their 1 lap trip around Puzzlaria?
Including the train each one is riding on, how many trains did DualAspect and Ross see on their 1 lap trip around Puzzlaria?
posted by Zaux at
6:22 PM
13 Comments:
If I understand correctly, new train sets off every 15 minutes, right? I think DualAspect saw 8 trains and Ross saw 12 trains, but I doubt it's that easy. I must be missing something :D
This post has been removed by the author.
Assume that trains have been leaving once every 15 minutes before Dual Aspect and Ross leave.
There will be 12 trains traveling counter clockwise at any given moment and 8 trains clockwise.
If you first imagine the counterclockwise trains to be stationary then you will pass all 12 on your trip. But since they are moving of course you will see even more of the trains. The counter clockwise trains will make 2/3's of a rotation during the a clockwise trains trip. So that's 8 more.
So 12+8+1= 21 trains
12 stationary.
8 extra for motion.
1 for counting itself.
I make that they both pass 20 trains each.....
The clockwise trains take 2 hours to lap, leaving every 15 mins. When the anti-clockwise train leaves the station there are 8 clockwise trains already on the way around the track. The anti-clockwise trains take 3 hours to lap, during wich time 12 more clockwise trains would have left on their journey.
The anti-clockwise train will pass all of the 8 clockwise trains that were on the track at the start of its journey, plus the 12 that left during its own journey time. Total - 20.
Similarly, when the clockwise train leaves there are 12 anti-clockwise trains on the track. During the 2 hour clockwise journey 8 more anti-clockwise ones leave the station.
The clockwise train will pass them all; 12 already on the track and 8 newly departed. Total 20.
However, if you are counting individual trains, rather than trains seen more than once, then the clockwise one will see 12 and the anti clockwise on will see 8, because the ones that depart from the station will be the same ones that have already been passed once before.
All of the above doesn't account for Ross and I counting the train that we are on as one that we have seen, and also ignores that we may see one extra train waiting in the station before we leave, so the answers could be 21 or 22 if you count these possibilities.
i think it is only 17 times
lets figure it out how
draw a circle imagine it as an analog clock
first train goes clockwise and meets a train each 15 min for a 2 hours trip around
so mark a dot for each encounter 8 in total
dot 1 : 0 second
dot 2 : 7.5
dot 3 : 15
dot 4 : 22.5
dot 5 : 30
dot 6 : 37.5
dot 7 : 45
dot 8 : 52.5
and for the second train going anticlockwise mark for each encouter 12 in total
dot 1 : 0 sec
dot 2 : 55
dot 3 : 50
dot 4 : 45
dot 5 : 40
dot 6 : 35
dot 7 : 30
dot 8 : 25
dot 9 : 20
dot 10 : 15
dot 11 : 10
dot 12 :5
so now you have the encounters
17 in total ( i think )
Hi Lapierto,
The trains would meet more frequently than every 15 minutes as they are moving in opposite directions towards each other with a combined velocity which means that they meet more quickly than each individual train is travelling.
To clarify: If you stood on a bridge over the track then a train would pass in each direction every 15 mins. If you rode a pedal cycle clockwise around the outside of the track you would meet an anti-clockwise train more often than every 15 mins, and a clockwise train less often than every 15 mins.
The same is true of the trains travelling towards each other.
Just read the question again and realised that it does say "including the train each one is riding on" I must learn to re-read the question before posting.
My answer then is the same as Neal's - 21.
This post has been removed by the author.
The problem states that the train each is riding should be counted.
no definitive correct answer yet
Ok, how about this then...
Clockwise (Dual Aspect) - 13 trains, 12 going the opposite way plus the one he is riding on.
Anti-clockwise (Ross) - 9 trains, 8 going the opposite way plus the one he is riding on.
There are 12 trains travelling perpetually around the anti-clockwise loop, the clockwise train passes all of them. It will pass 8 of them twice, but they will be the same trains as already seen so each is only counted once.
There are 8 trains travelling perpetually around the clockwise track which the anti-clockwise train will pass. It will pass all of them at least twice, some of them 3 times, but they will be the same trains so each is only counted once.
Am I on the right track (sorry for the pun!) now Zaux?
so your counting the both the trains leaving the moment you start and the moment you finish?
So 9 trains on opposite track if it were stationary. I had pondered the thought of meeting seeing 9 trains on that track instead of 8. Oh well I was close enough to be proud.
Dual ..
22 is right
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