The Factor/Multiple Game

As far as I'm aware, this is not a super well-known game. I first learned of it from Ian Stewart's ``Professor Stewart's Hoard of Mathematical Treasures.''

Description

Alice and Bob take turns saying whole numbers from 1 to 100 inclusive (let's assume Alice goes first) according to the following rules. (The upper bound of 100 can be modified to some arbitrary \(n\).)

Every number said (besides the first) must be a factor or multiple of the previous number said.
Once a number has been said, it may not be said again for the duration of the game.
Alice must begin with an even number as their first move.

If a player ever has no legal moves left, they lose.

e.g. \( 98 (A) \rightarrow 49 (B) \rightarrow 7(A) \rightarrow 77 (B) \rightarrow 11(A) \rightarrow 55(B) \rightarrow 1(A) \rightarrow 97(B) \)

In the above example game, Alice loses (from a rather unfortunate starting choice) because she has no legal moves remaining (97 and 1, the only valid factors/multiples of 97, have both been used).

Graphs

This game can be understood to take place a on a graph whose vertices are the whole numbers from 1 to 100 and whose edges are between \(x, y\) where \(x |y\) or \(y|x\). Some visualizations of this graph can be quite pretty. One way of rendering is to place vertex \(k\) at the point \((\cos(2\pi k / 100), \sin(2\pi k / 100))\). Doing so gives the following image.

If we change the upper bound \(n\) to something much larger, we see a finer version of the previous image. This particular one uses directed edges from factors to their multiples (purely for aesthetic reasons).

Strategies

One interesting remark is that the addition of the third rule is to prevent the game from being completely trivial. If we didn't have this restriction, then Alice need only pick a prime greater than 50 to immediately win. Such a chouce would force Bob to pick 1 as the only valid move, at which point Alice can choose a different prime greater than 50, ending the game.