Crooked Cannons:

For more of a challenge, make the fold at a slight angle across the page.

With a crooked fold, there are some parts of the paper where cannonballs can't reach. You aren't allowed to draw ships in these areas. After all the ships are drawn, fold the paper closed and make sure you can't see any ships peeking out on either side.

Space Battle:

Instead of drawing ships, make UFOs by tracing coins of different sizes. For example, each player might trace a dime, a nickel, and a quarter. Then flip a coin to see who goes first!

Double Fold:

Make two folds (one vertical, one horizontal) across the paper, forming 4 regions. Each player draws three ships in one of the top regions. (The bottom regions stay empty.)

On each turn, a cannonball will reflect over two folds. Draw a dot in the bottom corner under your own ships.

Then fold the paper along the vertical line and scribble to copy the dot to the bottom corner under your opponent's ships.

Redraw the dot over the reflection point. Then fold the paper across the horizontal line and scribble to copy the dot to your opponent's region.

Symmetry:

The fold in the sheet of paper acts as a line of symmetry for the cannonballs. By copying the pencil dot over the fold, players are creating a reflection of their cannonball over the line of symmetry. The kind of symmetry modeled by the cannonballs is called reflectional symmetry. Examples of everyday things that have reflectional symmetry are faces, butterflies, and letters like A and T. Can you think of more?

The Double Fold variation above reflects each cannonball over two lines of symmetry. The result is that the cannonball has rotated 180 degrees around the center of the paper. (You can imagine a 180 degree rotation by thinking of how far you have to rotate a piece of paper to turn it completely upside down.) This is called 180-degree rotational symmetry. Examples of everyday things that have rotational symmetry include windmills, pinwheels, and letters like N and Z.

Note that there are different kinds of rotational symmetry. In our game, rotating the paper 180 degrees (that is, turning it upside-down, or rotating it 1/2 of the way back to its original orientation) results in the cannonball's starting location overlapping with its ending location. But some objects don't need to be rotated 180 degrees to look the same. A pinwheel with 4 points would only need to be rotated 1/4th of the way around to line up to where it began. A pinwheel with 5 points would only need to be rotated 1/5th of the way around. A perfect circle can be rotated any amount and still look exactly the same!