Pi Day Riddle

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During the current Coronavirus pandemic, I consider myself fortunate to be able to work remotely, and to have (as of now) all family members safe and healthy. So while schools and most businesses in the state are closed, I have the luxury of being able to sit in the house and write a blog post about Pi Day.

Many are familiar with this riddle from middle school math class:

A man walks one mile South, one mile East, and one mile North, to end up exactly where he started. Where is he?

(In recent years it's become well-known as the Elon Musk interview riddle, as the CEO is known to ask this question to ascertain some job candidates' analytical thinking.)

I learned when I was a kid that there was only one place on Earth where he could possibly be: the North Pole. (This assumes that the North Pole is frozen over, as it is currently, but future classrooms may imagine the man is in a rowboat.)

But there is another possibility: Imagine that the man is a little more than a mile North of the South Pole. Specifically exactly (1 + 1/(2π)) miles.

He walks a mile South, to be 1/(2π) miles North of the South Pole.

Then when he walks a mile East, he goes in a circle around the South Pole, the circumference of which is one mile (C = 2πr, r = 1/(2π), C = 1.)

Then by walking a mile North, he is back where he started.

So that's an infinite number of points, around that circle of starting points 1/(2π) miles North of the South Pole.

Now, are there any other possibilities?

How about (1 + 1/(4π)) miles?

A mile South of that is 1/(4π) miles North of the South Pole.

In this case, when he walks a mile East, the circumference of the circle around the South Pole is 1/2 mile. (C = 2πr, r = 1/(4π), C = 1/2.) So he goes around twice.

Again, by walking a mile North, he is back where he started. So another infinite number of points around that circle of starting points 1/(4π) miles North of the South Pole.

This works if he starts at a point where he goes around three times, four times, etc. So he has to start at exactly (1 + 1/(N*2π)) miles North of the South Pole, where N is any positive integer.

So an infinite number of points around an infinite number of concentric circles around the South Pole, as long as the distance from the South Pole fits that formula.

(Of course, as N approaches infinity, the starting point approaches 1 mile North and the radius (and consequently the circumference) of the circle approaches zero. So the man is walking South a mile, then walking in many tiny circles just inches from the South Pole, then walking North. In practice, at that point it's imprecise, dizzying, and a bit silly. But still holds true, in theory!)

Are there any other possible answers to the riddle? I also welcome any other Pi Day riddles, thoughts or reflections.

Take care, and Happy Pi Day, everyone!