I came across some instructions for Euclid's Game which I don't think are correct:
Euclid’s game starts with two unequal positive numbers on the board. Two players move in turn. On each move, a player has to write on the board a positive number equal to the difference of two numbers already on the board; this number must be new, i.e., different from all the numbers already on the board. The player who cannot move loses the game. Should you choose to move first or second in this game?
The reason I don't think this is correct is that all other versions I've seen described claim that a multiple of one of the numbers should be subtracted from the other. Then the game repeats with just the two numbers remaining after this step. This makes the game mimic Euclid's Algorithm, which makes the name make sense.
The version given seems to end only when all numbers from 1 to the highest number have been listed.
For example:
Start: $1, 5$
player $1$ chooses $5, 1:$
$$1, 5, 4$$
player $2$ chooses $4, 1:$
$$1, 5, 4, 3$$
player $1$ chooses $5, 3:$
$$1, 5, 4, 3, 2$$
player $2$ loses as no new positive number can be made by subtraction.
What I'd like to know is:
Have I misunderstood the rules?
Are the rules incorrect for what is usually called "Euclid's Game?"
Will this version always produce numbers from 1 to the higher original number (or multiples of 10, 100 etc. up to the highest number) and if so, why?