## Thursday, February 10, 2011

### Love Triangle

Dating and Love - what do they have to do with math? This mischievous Valentine's puzzle shows that math may very well be the guiding hand in some romantic adventures.
One man had two lovers. One was a witty, fashionable journalist that lived in the North of the city, and another, passionate and eccentric actress that lived in the South. The man couldn't quite make up his mind about whom he loves more. So every day after work he walked to the Subway, went down the escalator and took the first train that would come. On his right were trains heading North, on his left were trains going South. Whatever train came first determined which lover he would spend the night with. He thought that because the trains go on a regular schedule, with the same frequency in both directions, he would end up spending about the same amount of time with each. He told each of the lovers that on the days he is not with her, he is travelling on a business assignment. All was going well, until one day the journalist confronted him claiming that he has been spending only about 2 days per week with her, while the actress started hinting that he has been crowding her life.

I do not understand, thought man. The trains are regular, I choose randomly, why do I get to one lover much more frequently than another?
What do you think?

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point. Please, explain your answer.

anne-marie said...

Congratulations to Kim for last week!

The trains go on a regular schedule and have the same frequency.Certainly that this man walks everyday on a regular schedule to the train station.(afterwork)
The probabilities that this man will take the same train going South are higher than taking the train going North.

Dennis (of Dennis and Katrina) said...

I can see three reasons it's not working out like he wanted:

1) In his city, the majority of the suburbs are to the South, which means the majority of the commuters live to the South. In the morning, there are a lot of people going North and in the evening, there are a lot of them going South. With more people going South in the PM, there are more trains going South more frequently to handle the load. Therefore, he's more likely to go South. His assumption on the distribution of north/south trains in the evening was wrong, so he didn't get the outcome he wanted.

2) It's a question of timing. He likely gets off work at the same time each day. It's working out that leaving at that time and walking to the subway puts him at the platform at such a time that the next scheduled train is likely heading South. The timing is close enough, though, that small, seemingly random things, like missing the crosswalk light, stopping to buy a paper, the phone ringing as he's walking out the door, etc., are delaying him enough that he misses the SB train and is hitting a NB train two days a week.

3) What does he expect? The work-week is typically five days, so if his assumptions are right, I'd expect 2 days one way and three the other. If he's crowding the actress, what's he doing on the weekend? Maybe whatever he's doing there is the source of the problem!

Tom said...

It's not a remarkable distribution, even several weeks in a row, to get only 2 out of 7 (heads, tails). Certainly that could happen and that sort of distribution is what destroys system players at a roulette table.

OR...given he is doing this after work, it is very possible there simply are more trains headed one way than the other (more Southbound trains, in this case). The problem says that the man *thinks* the trains are regular and in the same frequency in both directions. Man might be wrong about that.

And this fellow is not being very smart about it, after all. He could keep track. Dummy.

SteveGoodman18 said...

Let's assume a 5-day work week. Assuming the trains come at regular intervals, let's say they come every 10 minutes, although the exact interval doesn't matter. We also assume that the man's arrival time at the subway station is distributed uniformly.

If he is only going north about 2/5 of the time, then the northbound train might come 4 minutes after the southbound train.

For example, if the Southbound train came at 5:00 and the Northbound at 5:04, any arrival from 5 to 5:04 would send him north, but any arrival from 5:04 to 5:10 would send him south. Thus there is a 6/10 or 3/5 chance of heading south.

Wang said...

great puzzle! Even though the trains are on a regular schedule, one train comes to his station more often than the other because of his location. If he is closer to the north, there is a greater chance that the next train he takes is going north. The distance going north is greater than the distance going south and at any particular time, the chance that a train is going north > chance that a train is going south.

-lex- said...

If the northbound train comes always shortly after the southbound then it is very unlikely that the men hits the platform in the short time gap between the south and north trains. It is more likely that he arrives to the station during the long wait between the north train and the south train. Note that in the first unlikely case he visits his girlfriend on the North, and in the likely second case he makes the actress on the South happy.

These time gaps do not change, the trains follow a regular schedule, with the same frequency in both directions.

Annie said...

Since he was coming from work at about the same time every day he was not actually taking random trains. He would be taking the same train every day. Allowing for late trains or leaving work early would account for not taking the same train exactly every day.

Annie said...

Addendum: 2 days spent with the journalist could also be the weekend. If he consistently sees her only on weekends then he may be taking the train at the same time which means it is again not random.

Lynnet said...

His assumption that the trains are equally spaced is incorrect because if they were, he would be going to each ones house at the same rate. If we assume that the northbound trains come on the 1/2 hour and the hour and the southbound trains come on the 25 minutes and 55 minutes, there is a five minute offset between the trains. If he arrives at the same time as the train he misses it and takes the next train. This means that out of one hour, there are only two times he could arrive and make the northbound train.

Maria said...

Annie & -lex- welcome to the marathon!
Wang - good to have you back - we missed you for a few weeks.
Dennis - please tell your other half (Katrina) that we miss her as well.

Everyone who posted their ideas above has some valid arguments and therefore gets a puzzle point: anne-marie, Dennis, Tom, SteveGoodman18, Wang, -lex-, Annie, Lynnet.

While the weekend answer and man arriving at about the same time every day answer definitely make sense, here we assume that man works throughout the weekend (to support two lovers!) and that he gets out of work randomly at the different time every day.

The beautiful mathematical solution is in the train schedule and some of you got it absolutely right. While trains in both directions go with the same interval, train schedule is arranged in such a way that after every North bound train leaves, one has to wait 5X mins till the next South bound train. And after every South bound train leaves one needs to wait 2X for the next North bound train. This makes it 5/2 times more likely that when man comes he will be waiting for the South bound train and will go to his South lover.

For example, if the Southbound train came at 5:00 and 5:07. And the Northbound at 5:02 and 5:09. Any arrival from 5 to 5:02 would send him North, but any arrival from 5:02 to 5:07 would send him South.

Happy Valentine's Day everyone!
Another dating puzzle next week.

Meanwhile - enjoy the Valentine's Math Quiz

Maria said...

Wow - Lynnet just earned her 10th puzzle point! We are going to wrap her in a puzzle next week. But I have promised a dating puzzle... Perhaps we will wait with another dating puzzle for a few weeks :)

Bean said...

I'm going to post a "late entry" just to join the group...I thought I submitted from my iPod Touch in the middle of First Grade Movie Night at school last night, but it didn't go through!

If the trains are equally frequent and regularly spaced, BUT the spacing between arrivals is not split evenly, he will more likely to catch one train than the other. So, for example, this could happen: Southbound train; 2 minutes pass; Northbound train; 5 minutes pass; Southbound; 2 min; Northbound; 5 min, etc. Then the odds of the next train being Southbound would be higher (5:2).

Maria said...

Hey Bean - we'll believe you. And your answer is absolutely correct. I spent my Friday night as well as this whole weekend on my son's swimming competition.