Why doesnt the product game have 13 on board?

You're likely referring to the classic game "Product Game", sometimes called "Dots and Boxes", or "Dots & Boxes".

The reason there isn't a 13 on the game board is due to the way the game is structured. Here's why:

* Grid Structure: The board is a grid of dots, and players connect the dots to form squares. The number of dots on the board determines the potential number of squares.

* Square Formation: To complete a square, you need to draw the fourth and final line. This means that the number of squares you can create is always one less than the number of lines you can draw.

* Number of Lines: The number of lines you can draw is determined by the number of dots on the board.

* Mathematical Relationship: This leads to a simple mathematical relationship: Squares = Lines - 1

Therefore, the number of squares on a Product Game board is always one less than the number of lines that can be drawn. This means that if there were 13 squares on the board, there would have to be 14 lines, which is impossible with a standard Product Game board layout.

Common Board Sizes: The most common board sizes are:

* 2x2 grid: 4 dots, 4 lines, 3 squares

* 3x3 grid: 9 dots, 12 lines, 11 squares

* 4x4 grid: 16 dots, 20 lines, 19 squares

So, the absence of a 13 on the board is a direct consequence of the game's rules and the way squares are formed.