The Grey Cells - Exercise !

Here's one from Martin Gardner:

A commuter is in the habit of arriving at his suburban station each evening exactly at 5 o'clock. His wife always meets the train and drives him home. One day he takes an earlier train, arriving at the station at four. It's a nice day, so instead of waiting for his wife he decides to walk towards home, using the route his wife takes. They meet somewhere on the way. He gets in the car and they drive home, arriving there ten minutes earlier than usual. Assuming she always drives at a constant speed, and that on this occasion she left just in time to meet the five o'clock train when it arrived, can you say how long the husband walked before he was picked up? (By his wife, that is )

 

When you ask "how long the husband walked" - that means a distance to me, like "hvor langt manden gik" in danish. 

But if the same can mean "for how long (time) the husband walked" - then of course, you are asking about a period of time, and that makes it somewhat easier.

How's this:
They get home 10 minutes earlier than usual - meaning the total driving time for the wife is 10 minutes less than usual, 5 minutes in each direction.

Then she must have picked him up 5 minutes earlier than the usual 5 o'clock. She has already earned the first 5 minutes out of the 10 minutes saved, in order to reach him. So she picked him up at 4:55 and as he got off the train at 4:00, he must have walked for 55 minutes.

 

 

 

 

 

Northman, in English if I want to know the distance someone walked I would say 'how far did he walk' not 'how long'.

 

It's an interesting point though. I would say 'how long is this string' looking for a distance. I'm having to think hard to formulate a rule for when to say 'how far' and when 'how long'.

 

How far is it to New York from Luxembourg? About 3,000 miles.

How long is it to New York from Luxembourg? About 7 hours.

 

 

and yes - size is area.

And I'm asking for a polygon the same size as a 2x2 matches - but using all 12 matches.

 

 

 

 

I proposed this solution in a PM, however, Northman suggested that there was a more 'intuitively obvious' solution.  By that I assumed that the match sticks would be arranged at right angles or parallel to one another so that one could 'see' the four 'squares' making up the total area (as one can see the five in the righthand figure originally posted).  However, I can not see how, as the smallest area I can see being contained by the 12 match sticks, given the additional constraints is 5 - i.e. a 1 x 5 rectangle, or other forms topologically equivalent to it.  So I provide my solution, however, I will concede to anyone who can construct such a polygon with area 4, as I have described with the additional constraints.

 

p.s. I'm not sure how to post an image that I have 'constructed' myself, but is not on a website.