## Pawns

Consider an infinite checkerboard divided in two with an infinite line lying along the x-axis, as depicted below:

Let’s zoom in a little:

We will play a game on this board. In this game, you start with some pawns under the horizontal line. For example, here is a potential starting configuration:

In each turn you can make a pawn jump over one of its 4-connected neighbors, thereby killing it (removing it from the board). The pawn’s movement is demonstrated here:

Will become this:

When the game starts, all the pawns are below the line. You can move the pawns as per the rules above, and you get points depending on *how far above the line* you can go. For example, here is a starting configuration with 2 pawns and the first move:

Resulting in a game ending with 1 point:

It is obvious that in order to make a pawn cross the line (or move at all), at least two pawns are needed, so all games with a single pawn will end in a score of zero.

How many pawns are needed in order to score 2 points? As is demonstrated below, 4 pawns are sufficient, and indeed this is the minimum (check that you see why 2 and 3 pawns cannot do it).

Now for the riddle. How many pawns are needed in order to score 5 points?