Start with a collection of paychecks, from $1 to $12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose.
Then you choose again from the remaining paychecks and so on.
The tax collector must receive payment after every move.
If you have no moves that give the tax collector a paycheck, the game is over and the tax collector gets all the remaining paychecks.
Is it possible to beat the tax collector in this $12 game?
If so, what is the maximum amount you can get?
If not, show why not.
Bonus: same game, starting with 48 paychecks ($1 to $48).
Credit goes to Daniel Finkel of NYT, whose puzzle I have slightly modified.