This is allegedly Elon Musk’s favourite question
that he often asks in his interviews.
Suppose you’re standing at a place on Earth
such that when you walk 1 mile south,
1 mile west and then 1 mile north,
you end up at the same position.
Where could you possibly be?
And how many such points are there on the Earth?
You can pause the video and try to answer it.
Ready?
Let’s start.
Some of us may think it’s impossible.
It is impossible, if we try to represent the problem on a piece of paper.
1 mile south, 1 mile west and 1 mile north.
We land up one mile to the west of where we started right?
But is the earth flat?
According to many computer generated models,
the true shape of the earth is like that of a potato!
But to understand the solution to this problem,
we will assume that the earth is a large spherical ball.
To understand the directions north, south and west,
let’s draw the lines of latitude and longitude.
Now, the direction north means to travel
along a line of longitude towards the north pole,
and vice-versa for the south direction.
The west direction means to travel from right to left
along a line of latitude.
Now that we understand the directions mentioned
in the problem in a better manner.
Let's try to attempt a solution to the problem
and take any random point on Earth.
As we traverse 1 mile south, then 1 mile west and 1 mile north,
we will see that we don’t end up at the same point.
But what if we were at the north pole?
Try repeating the same scenario for north pole.
If we go one mile south, then one mile west and then one mile north,
we see that we end up where we started from.
That is the north pole.
So the north pole is one such point
that satisfies the conditions of the problem.
But have we solved the problem?
Not quite.
Can you think of more such points?
There is a class of points we haven’t included in the solution.
To find them, let’s understand the problems core movements and logic.
We see the directions north and south can cancel each other out.
If we go one mile south and one mile north, we reach the same point!
However, the distance we travel in the west direction
is not allowing us to do so.
So, if we’re able to cancel the net effect of motion in the west direction,
then we’ll always reach the same point we started from
at the end of the path.
So, what would be a way to achieve this?
How do we cancel the net effect of the west direction?
Consider a circular path of circumference one mile,
north of the south pole.
Let’s call it C1.
Now if we take any point on the circle, and travel 1 mile westwards,
where will we reach?
You are right!
Back to where we started.
So if we are at this point, and travel one mile westwards,
we come back to the same point.
With this, we have nullified the net effect of the westward movement.
Now can you try finding more points on Earth
that satisfy the condition mentioned in the riddle?
Let’s take a point one mile north of this circle
and traverse the mentioned path.
As we travel one mile south, we reach the circle C1.
And when we travel 1 mile west,
we complete a loop on this circle and reach the same point
on the circle where we started from.
And finally when we move back one mile north,
we end up at the same point where we began our journey from.
So, this circle helped us to find another solution to the puzzle.
This point here is another solution!
Did this help us find just one more solution?
Surprisingly, it helped us find infinite MORE solutions.
If we look carefully, every point on this shaded circle
will be solution to the puzzle.
Doesn’t matter which point we take, and traverse the directions mentioned,
we will reach back at the same point.
Is that it?
All points on this circle and the north pole are the solutions to this riddle?
Can we find even more solutions?
Yes we can!
We are able to find solutions because
we are able to cancel the effect of motion in
the west direction by traversing in a loop.
However, we could have taken TWO rounds around a circle
half as long and reached the same point.
Did you understand what I am trying to say?
Let’s consider the circle with circumference half a mile like this,
and call it C1 half.
Now let’s take one point north of this circle.
We travel one mile south first to land on the circle C1 half.
To travel one mile west, we will need to travel 2 loops around this circle
and we will be back at the same point!
And then we travel one mile north to reach the same point where we started!
Now if we take any point on the circle 1 mile north of this circle,
we’ll see that it is a valid solution to the puzzle.
Again, infinite points!
So what does that tell us?
It tells us that we can cancel the effect by travelling in 3 loops,
4 loops and so on...
So for the circles with circumference 1 over 3 miles,
1 over 4 miles and so on up to one over ‘n’ miles,
we find infinite solution points 1 mile north of them.
This completes our solution to the riddle.
There are infinite solutions to the problem.
These solutions can be mentioned as north pole
and the points on the circles, 1 mile north of the circles
in the southern hemisphere, with circumference ‘one over n’,
with ‘n’ being a natural number.
