January 02, 2011

A walking problem paradox

The Problem

A question goes like this:

"Michelle and Mike begin walking and go 5 miles due north.
Their friend Devrae begins at the same point and goes 12 miles due west."

The values are:

a) shortest distance between Mike and Devrae      b) 13 miles.

You have to answer if:

a>b or    a<b or   a=b or   cannot be determined.

The answer was said to be a=b with a nice 5-12-13 triangle and Pythagoras rule.

[caption id="" align="alignnone" width="248" caption="Is that right?"]Is that right[/caption]

Is that really the right answer?

So, is it? I'd like to argue not. From the question we can only assume M, M and D are humans, and walking around on some land somewhere, on this planet. Let's say they started walking from a point on the equator. MM go north along a longitude, and D left on the 0 latitude.

On a piece of paper the distance would be 13 miles, but given the curvature of the planet, the distance between the starting point and M/M would be < 5 miles.. because they walked  along a curvature, and the shortest distance (the chord) would be say 4.9 miles. Same goes for D.. the distance from start point is 11.9 miles. Thus the shortest distance is possibly less than 13 miles. So the answer could be a<b.

Then there's the ambiguity of the directions. Is west the direction of the sunset, or is it simply a reference to the direction left? If the former, the answer could goto cannot be determined because the latitudes are equidistant, but longitudes are farthest apart at the equator, and next to each other at the poles. So the answer would vary depending on the starting point (and it is possible that it is always less than 13 miles, due to the curvature of the planet).

What is that I hear? "Anil, you didn't think of the mountains and the cliffs! what if they were in the way?". I think a round flattened planet is a good approximation to make in walking problem (specially one that goes into miles) Think of the problem in terms of 1200 miles, and the distances and curvature become more apparent. Heck, what if the distance is > circumference .. M may end up in the same damn place again.. being only ~12 miles away from D!

So aptitude question writers, please refrain from using walking and driving in your distance questions, and east and west for directions. May I suggest flying spaceships, and left and right.

Of course, there's the whole warped space thing to consider then. ;)