Consider the following single player game:
The battlefield is a given (finite) collection $A$ of natural numbers. Example: $\langle 1, 2, 3 \rangle$
For every pair $x, y \in A$, whenever $x > y$, $x$ may be replaced with $x - y$, which I call reduction. A reduction constitutes a turn in the game.
The game ends when no more turns are possible.
At any point into the game, there would usually be a selection of turns available to choose from, so there will be many different strategies for playing the same setup, unless it is trivial.
example 1 $ \langle 1,2,3\rangle \to \langle 1,1,2 \rangle \to \langle 1,1,1 \rangle $
example 2 $ \langle 1,2,3\rangle \to \langle 1,2,2 \rangle \to \langle 1,2,1 \rangle \to \langle 1,1,1 \rangle $
proposition 1 We can see that a game on any battlefield must eventually end, as:
- Numbers are always made smaller.
- But can never be made smaller than 1.
proposition 2 A game ends exactly when all the numbers are the same.
Otherwise there is a possibility for a turn to be made.
corollary We may associate an ending with an "outcome" number.
proposition 3 Order of numbers on the battlefield does not matter.
Does a game on any battlefield only have a single outcome, regardless of strategy?
Can outcomes be predicted without actually playing?
Is there a strategy that ends the game on any battlefield the fastest?
Does repetition of numbers on the battlefield matter? In other words, may the outcome change if the battlefield is converted to a set?
I can see how the outcome (assuming its uniqueness) is bounded above by the smallest number available and, furthermore, by the remainder of the division of any two numbers present on the battlefield that do not divide evenly (since integer division is a series of subtractions). For example: if there is $1$ on the battlefield, the game will necessarily end with $1$; if there are $13$ & $27$, the outcome is again $1$. But a remainder obtained at some point in the game may possibly in turn be used to obtain an even smaller remainder! I have no idea how to pinpoint the end of this sequence of ever smaller remainders.