Question

Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending positions of the "leaper". Will a frog ever occupy the vertex of the square which was originally unoccupied?

Answer 1

Let us develop some terminology to help us better understand the problem.

We will say that the co-ordinate pair is a lattice point if both and are integers.

For example: is a lattice point while is not.

Let us suppose, we have a unit square with vertices , . and .

Without loss of generality, let us assume that the three frogs are at the vertices , and respectively.

If not, we can always rotate the square to do so.

Another terminology! ... being even and being odd, observe that , , and are of the form , , and respectively.

Claim is that the vertex is never occupied.

In fact, we will prove something stronger that we never reach any point with parity..

So, where do we start?.. It is only fair that we start from the beginning and use what is given to us.

Starting we have, before any leap is made, the occupied vertices are , and .

What happens after the first leap made by anyone of the frogs?

Say, the frog at leaps towards the frog at . That means is the mid-point of starting position and the ending position of the leaper, say, .

Solving, we get .

What did you observe? That the ending position is a lattice point.

Also, is . And so is the ending position .

So, it leads us to think that given any point of time, the positions occupied are lattice points with one of the parities among , and .

We will prove the same using the method of infinite descent.

i.e., for any positive integer , the occupied vertices after leaps are lattice points with one of the parities among , and .

We have shown it for above.

Suppose that the result is true for .

i.e, the occupied vertices after leaps are lattice points with one of the parities among , and .

So, what happens when a new leap has been made?

Suppose there is a frog at a point which wants to leap towards another frog at a point .

By previous result, and are lattice points with one of the parities among , and .

Now, where does the frog land?

Say, it lands at the point where are only real numbers to begin with. We do not know if they are integers.

i.e., is the mid-point of and .

and .

and .

That means are integers.

Also, observe that

So, if , or , then would also have to be , and respectively..

That means even after leaps, the occupied vertices are lattice points with one of the parities among , and .

Since is any positive integer, we have that the occupied vertices are lattice points with one of the parities among , and .

That means the vertex is never occupied. This finishes the proof.