Reach for the stars

Welcome to Stars! The object of the game is to create a certain number of stars by adding or removing stars from your current collection. You always start with a small positive number of stars. On each of your turns you can apply one of the available moves to your current collection to construct another collection (which becomes your new current collection). The list of available moves is different for each level of the game. It may require some study and investigation on your part to come up with an efficient strategy for beating this game at the higher levels of difficulty.

BONUS Challenge: Can you determine all of the possible collections you can construct from your starting collection using only the given moves?

Play Stars!

Game Components

As described in the Introduction, this game illustrates the basic features of a formal proof system.

The Math Behind the Fun

This game is related to the branch of mathematics known as number theory. In particular, winning a particular game is equivalent to finding some way of expressing the difference between number of stars in the goal and the number of stars you start with as a linear combination of the given move numbers where the coefficients in the linear combination must be nonnegative integers. In some levels and instances of the game you can produce any positive number of stars. In others you cannot, in which case it is an interesting mathematical question to determine exactly which goals can be reached from the starting collection in a particular game. Fortunately, the game only picks goals which can be achieved, so you can be confident that there is a way to win the game, even if you can't find it.