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Tax Collector (Posted on 2015-05-26) Difficulty: 2 of 5
The Tax Collector game is played like this:
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.

See The Solution Submitted by Ady TZIDON    
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Some Thoughts Answer / no solution | Comment 5 of 6 |
Starting with n paychecks ($1 to $n)
yields the following sequence which I did by hand up to 12
0,2,3,7,9,15,17,21,30,40,44,50

This was enough to look up the sequence
https://oeis.org/A019312
which implies the solution for the n=48 problem is 734 but does not indicate the order of the chosen checks.

(Obviously 47 is chosen first, 48 is very likely chosen last.)

  Posted by Jer on 2015-05-26 09:51:14
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