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 Tax Collector (Posted on 2015-05-26)
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 No Rating

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 Thoughts. | Comment 1 of 6

Is it possible to beat the tax collector in this \$12 game?

Yes.

Take 11 first. Its sole divisor is 1. T takes 1.

After this no more 'prime wages' can be taken.

Continue with 8. T takes 2 and 4.
Continue with 9. All the factors of 9 are 3, so T takes 3.
Continue with 12. T takes 6.
Continue with 10. 5 divides 10 and is available. T takes 5.
The game ends; T takes 7.  1 to 7 = 28, 8 to 12 = 50

Edited on May 26, 2015, 9:05 am
 Posted by broll on 2015-05-26 08:51:38

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