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December 2000

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Question 1: 50 minutes.
The wife saved twenty minutes of commute time, meaning (subject to assumptions) that driving from the depot to where she picked up her husband would have taken ten minutes. So she picked him up at 5:50, meaning that he had been walking for fifty minutes.

Selected Answers to Question 2 (Q1 will be posted on Dec 29th)

A short essay from Jonathan Hayward
Part 2: There are a whole host of assumptions, most of them boiling down to treating the problem as a mathematical abstraction, and the concomitant cultural assumptions that go with story problems.
These are not necessarily bad -- making such assumptions greatly reduces cognitive strain.  If a high school physics teacher asks the students "Assuming a throw is made from ground level, what angle should a baseball be thrown at to go farthest," and then docks points when students do some trig and answer 45 degrees (instead of about 30 degrees -- the human arm can throw a ball faster at closer to horizontal angles, and the added speed of an athlete's strongest 30 degree throw makes up for the trigonometrically suboptimal angle).  Refusing to make these sorts of assumptions is, in mathematical problem solving, what refusing to make a willing suspension of disbelief is in literature.  Throwing down _The Lion, The Witch, and the Wardrobe_ because it portrays magical portals and talking beasts as if they were as real as children and mansions, is not a mark of a perceptive and discriminating mind; it is a refusal to enter a cultural arena by the rules that most players do not need to have written down.  If there is an assumption to make explicit, I would question the implicit assumption behind your statement of part 2, namely that assumptions are basically bad, a mark of imperfect thought, and a source of error.  They can be, but need not be; I assume a number of things (for example, that your problem means what my culture has taught me a document like your web page says it means -- namely, a challenge, one appropriate response of which is to think and e-mail you an answer; that you will not sell my e-mail address to spammers or otherwise make me regret a good faith response; you prefer correspondence to be in English or perhaps another language that can be easily translated to English.; other assumptions that I cannot consciously think of for the same reason a fish can't think of water).  My making of those assumptions is not bad thinking, nor is it likely to get me into trouble -- at least not as likely as carefully avoiding making assumptions.  If I carefully avoided assumptions like those above, that would open the door to sending you a solution in a language I created, or a book review for whatever I'm reading at the moment, or my computer's newest core dump.  I believe things go better if I make assumptions and e-mail what I have been taught to assume are acceptable responses to your web page.

More Answers:

The natural way to solve the problem gives an answer of 50 minutes. Some of the more realistic objections to this procedure: Travel towards the station is slower than away from the station. The wife normally arrives at the station earlier than 6:00.

Some of the wilder objections follow.

Assumption: Euclidean space time. Now if we're being strict, I think the acceleration and the return trip around a spinning body means there will be some attoseconds lost here and there.

The up and down speeds are the same.

Hit a deer because it was closer to twilight. (Note: these problems are posted from Westchester County, NY, where this is a problem.)

She drives at a constant speed and takes infinitely small time to reverse the car.

The wife always takes the same route going *to* the depot as she takes coming back *from* the depot. This assumption is *not* warranted by the fact that it was a daily routine. In fact, it is not warranted at all: one-way streets and varying traffic patterns are just two of the many factors that could cause someone to habitually take different routes going to and from a particular place. Since the husband did indeed meet his wife along the way, this assumption must be correct; but if she took a different route going to the depot, one her husband was unaware of, he could have missed her as he walked back along the wrong route.

At 5:40 these days it's dark - was he an amateur astronomer? Maybe he saw a UFO... Maybe she saw a UFO?

The car trip always takes the same amount of time, and today was no exception. Again, this assumption is not warranted. Traffic jams, stop signs, construction work requiring a lowered speed limit, any number of things could cause a trip to be slower than usual.

At the point where the wife met her husband, turning the car around was not a problem. Like the previous assumption, this assumption is not warranted, but is necessary if the puzzle is to have a solution. The solution assumes that turning the car around took essentially no time; but if the husband and wife met at a place where turning around was near-impossible (a bridge, for example) and had to drive for a couple of minutes before they were able to turn around, then the solution would be substantially changed.

The wife left from home(as usual) at the usual time.

The home-depot average speed is the same with the depot-home one.

The average speed of the car journey for the return part is not affected by the fact that it is undertaken at an earlier time.

An example of what could go wrong, is that because the return journey is taken at a different time, they might be held up at a level crossing - train and road at same level, controlled by automatic gates.

The assumptions I made that were not stated are that the man's wife reaches the depot everyday at 6:00 sharp. She can as well come early!

Also, all events like the man getting into the car, the man's wife reversing the car, etc. were all assumed to happen in zero time on all days and on this day too, which is against the basic law of nature that no event can happen in zero time.

Wife actually picks man up, and after picking him up refrains from screaming at him maniacally for departing so drastically from the routine without bothering to inform her.

The speed of car MUST be the same all the way especially consider two journeys his wife took. We should ignore the factor may be she may drive the car faster in the journey toward the station.

Somewhere between leaving the Depot and meeting his wife he may have taken a return trip to anywhere on a close to light-speed traveling device. So we would face relativity issues here. Whose time would we use to count the minutes, his time or her time?

He could have made some pauses on his way home, he could also have run (instead of walk) for some minutes, or he could have gotten a short ride for a segment of the path. So technically speaking, he may have "walked" for a number of minutes much smaller than 50.

They stopped to yak about why he's home early.

Though his wife was driving the car when they met, maybe from X to home *he* was the driver. The statement of the problem only says he "got in the car" without further details, and then "they" drove home. He may drive faster than she does, and that could explain the difference with respect to the "usual", in which case she is the driver.

This day the car may have just been a faster one, or incidentally this may have been the only day in the week when there is less traffic, which always allows to get home faster "than usual".

Maybe they usually stop for coffee, and they just didn't this day. Maybe they usually pick up someone else on their way home (e.g. daughter and son from school) which drives them away from the shortest route between the Depot and home adding some time to their returning. Their children could have been away camping this day, so they managed to arrive home earlier "than usual" because of that, not because of the walking of the father.

The distance his wife traveled in the last 10 minutes every day to the depot is the same as the distance she traveled in the first 10 minutes after leaving the depot. This could be wrong. For instance, he could have met her at 5:55 p.m., but because it took her a long time to get out of the station (as lots of people were leaving), they save 15 minutes on the trip back home.

The time it takes for her to travel from the point at which they met to their home is the same regardless of the time they started. But the man may have met her at 5:10 p.m., but they could then get stuck in one-way com mute traffic, and get home only 20 minutes before usual, because traffic is much heavier around 5 than 6 o'clock.

Even the placement of their home (as they could have moved to another house closer to the hometown depot between the time of the two cases compared.)

The time it takes her to reach the depot from home is the same time it takes her to drive from home to the depot (e.g. the road is not uphill one way).

What if there were an accident on the road on her way there - traffic would be backed up and it would take her longer than normal to reach the station, but unless there was an accident in the other direction as well, she would get home in the normal amount of time.

It is also assumed that it takes no time for the wife to pick up her husband - i.e. slowing down and stopping to pick him up and then accelerating again to her normal speed after he gets in the car, and also that it takes no time for her to turn the car around - i.e. she can make an immediate U-turn without having to get off at an exit or turn into a gas station or something.

Commuters on their way home from work clog the outbound lanes, but not the incoming ones.

The man takes [ more time | less time ] getting into the car when picked up after walking than he did at the station.

It could have been the office Christmas party, he returned the worse for festive consumption and forgot to start walking or spent some time in non-business pursuits with an attractive intern. He might have caught his tie in the door, causing him to adopt an unlikely pose in the passenger's seat and attracting the unwelcome attention of a traffic cop. He might have been unable to negotiate the door handle successfully (it was a GOOD party).

The journey home might not be time-symmetrical (very likely) - Santa's sleigh double parked causing long queues on the way home. The journey is not distance-symmetrical, because of one-way systems,

Inclement weather, especially snow or fog (easy to walk in, hard to drive in), and most especially if it wasn't consistent.

Travelling at a different part of rush hour, so goes faster or slower. (Encounters the school bus returning kids from after-school activities)

How long, in minutes, did he sit on the bench for a breather?

If they usually stop at a bar together before going home, then it would depend if that bar is en route, or if the man already passed it (and stopped or not, who knows) on his walk home.

The wife actually leaves from the home when she goes to pick him up. If not, and he knows whence his wife is coming to pick him up (not directly from home) he might walk in that direction (say the hairdresser's) to meet her and get picked up. Again, we could not make any definitive statement about how long he might have been walking.

That his wife doesn't drive recklessly while he is not in the car (so her speed is different in different directions).

All times mentioned are assumed to be all PM or all AM. What if, on usual days, the man reaches the depot at 6:00pm. And on that special day, left the office at 4:00 AM (don't ask me why) and reached the bus depot at 5:00 AM ? And then he walked from 5:00 AM until he met his wife who was expecting to pick him up at 6:00 PM ? The man must have walked for 12 hours and 50 minutes! (of course one assumes that he didn't reach home even after so much walk; maybe he took long breaks;)

The home they go to is unique. What if, that was a Friday and they go to their country home on Fridays and the country home is closer to the bus depot ?

No daylight savings time correction has been applied. If it were applied at the place where the bus stop is located and not at the place where their home is situated (or vice versa), results would be different.

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