And the "!" sign means "factorial" in math.

like 3! = 3x2x1, 5! = 5x4x3x2x1, etc.

It was meant to convey the notion that there are more positions in the game of Amazons than the game of chess.

There are 26 trillion ways to have the board completely filled. Think of how many positions are possible along the way!

Today I was curious to know "how big" this game is when compared to chess. I decided to start by counting the number of positions where there is a definite loss.

There are 100 squares on the board, and when 92 are occupied, it must mean white has lost since black made the last move.

How many positions like that are there?

Well there are 100!/92!8! ways to have 92 arrows shot, and 8!/4!4! ways for white to occupy the remaining 8 empty squares, and 4!/4! ways for black to fill in the last 4 squares for each of these positions.

100!/92!8! =

100x99x98x97x96x95x94x93/8x7x6x5x4x3x2 =

186,087,894,300

8!/4!4! = 140

4!/4! = 1

So the total is:

26,052,305,202,000

Over 26 trillion ways to lose if you have filled every square on the board.

That would change nothing. If your position is so bad that you pass, your opponent just passes as well. Then, next time you pass, you are done.