Multiplying Single Digit Numbers to Get Product >1000

Question

This is yet another Alice and Bob problem.

Alice and Bob are playing a game on a blackboard. Alice starts by writing the number $1$. Then, in alternating turns (starting with Bob), each player multiplies the current number by any number from $2$ to $9$ (inclusive) and writes the new number on the board. The first player to write a number larger than $1000$ wins. A sample game might be (each number is preceded by "A" or "B" to indicate who wrote it):

{A1, B3, A12, B60, A420, B1260; Bob wins.}

Which player should win, and why?

I am assuming this problem uses optimal strategy on both sides, but as there are several ways to win, I'm not sure how to prove who should win.

All I do know, is that, if the number after your operation exceeds 1000/18, but isn't greater than 1000/9, it's a guaranteed win for you.

Can someone provide me with a solution? Thanks

I also apologize for the lack of a proper tag, as I didn't know a phrase that would correctly encompass this particular scenario/game.

Answer 1

bob can pick between 2 and 9. 7,8 and 9 dont work as Ross Milikan pointed out as any number more than 56/9 loses. For 4 and 6 which from bobs perspective are the winning numbers. if he picks between 4 and 6, Alice must pick between 8 and 54, so Bob can then pick between 56 and 111. Whatever Alice then picks, Bob can multiply by 9 and win, so 4, 5 and 6 are winning. 2 and 3 are not possible for Bob if Alice knows what she is doing as she can then make 4 or 6 on her next turn and use the reverse logic against Bob

Answer 2

As columbus8myhw says, $[112,1000]$ wins for the next player, so $[56,111]$ wins for the previous player, $[7,55]$ wins for the next player, $[4,6]$ wins for the previous player, so Bob should start with $4,5, \text{ or } 6$