Position of Camp

Here is a problem by J.A.H. Hunter.

'What a way to spend Christmas Day,' John muttered, screwing up his eyes against the blazing sun, 'but I guess there's some reason for it.' He figured he's trudged five miles due south since leaving camp that morning, and his briefing had been very clear. So now he set up the little radio beacon, and started off again, heading due east. The trek seemed endless, always east and still east until at last he reached his objective after a further 11 miles. From there on it was easier going, only a matter of five miles due north back to camp. What was the position of that camp?

Please give the answer and the method to get the answer.

Peter Heichelheim

Spoiler

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North Pole

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There are infinitely many possibilities for the position of the camp. To simplify matters, I will assume that the camp is on Earth, and that the reference to Christmas Day and blazing sun is intended to turn our thoughts to the southern hemisphere. Even with those two constraints, there are /still/ infinitely many solutions.

Let the camp be d miles north of the south pole, where d > 5.0 (because he travels five miles south. If the puzzle had set 'sets out toward the south and goes in a straight line for five miles', we'd have a whole new class of solutions). We have a valid solution in all cases where walking 5 miles south puts John on a latitude whose circumference is exactly 11/k, where k > 0. Thus, 11/1, 11/2, 11/3, 11/4, 11/5, etc. Note that, for every one of these solutions of d (of which there are an infinite number), there are an infinite number of places on the camp's latitude where that camp might be placed. Thus, we have an infinity of infinities, /after/ adding two seemingly restrictive constraints.

The method for getting the answer? Why, to read lots of books by people like Ian Stewart and Martin Gardner, of course! (I don't think this exact problem is in any of them, but Gardner certainly mentions closely related puzzles.)

I was very tempted to put our hero John on the /moon/, but this seemed overly exotic.

Another solution springs to mind: he's more or less on the equator, and the camp is a boat which drops him off on the northern coast of an island, and moves eleven miles east so as to be in the right place to pick him up again, but of course this is less faithful to the spirit of the original question, since it introduces extra bits in the manner of a lateral thinking exercise.

Or, Slightly more than 5 miles away from the South Pole.

The answer couldn't possibly be 'North Pole'. On Christmas day, the North Pole experiences 24 hour darkness, yet in the puzzle there was a blazing sun!

This clue indicates that it is a 'near South Pole' solution

Wouldn't that mean that he'd pass the camp k times?

The north pole.

Method used: common sense.

Does common sense say that you screw your eyes up against the blazing sun when it is night?

Yes. The puzzle doesn't forbid this. It says 'at last he reached his objective after a further 11 miles' but it doesn't say he reached it *for the first time* after a further 11 miles.

There is no sun at the North Pole on Christmas Day...

Therefore the camp must be about 6.75 miles from the South Pole .

And, after all, it also says that 'his briefing had been very clear'. Perhaps he was required to walk the circle of latitude exactly 4 times.