Each bug moves to an adjacent square. Prove that at least 1 square will be empty.

Skip to content
#### Ladybug Math

####
One Response

### Leave a Reply

Each bug moves to an adjacent square. Prove that at least 1 square will be empty.

Paul PadillaWLOG put the bugs on a 3×3 checkerboard with 4 black squares & 5 white squares.

Each bug starts on either a black or white square.

If each bug moves to an adjacent square, the color of the square that the bug is on will switch.

In this case, the 4 black square bugs will move to occupy 4 white squares.

Since there are 5 white squares on our board, there must be at least 1 white square with NO bugs on it.

We can also come to another conclusion:

The initial 5 white square bugs will move to occupy 5 black squares.

Since there are only 4 black squares on our board, there must be at least 1 black square with 2 bugs on it.