Merry
Tax
Day,
fellow
law-abiding
citizens.
As
national
observances
go,
Tax
Day
could
use
a
rebrand.
Some
traditions
to
make
it
more
fun
for
the
kids.
We’ll
adorn
our
tax-mas
trees
with
1040s
and
W-2s.
Children
will
hang
their
stock
portfolios
from
the
mantle.
The
naughty
dependents
get
penalties
while
the
nice
ones
get
deductions.
Charitable
families
will
take
a
day
to
serve
soup
at
the
local
tax
shelter.
We
already
have
a
bearded
mascot
for
the
occasion.
Who
wouldn’t
want
Uncle
Sam
to
slither
down
the
chimney
with
surprise
audits?
The
most
wonderful
time
of
the
fiscal
year.
Around the 1970s, mathematician Diane Resek invented “The Taxman Game” as a teaching tool for young students to practice arithmetic. The goal of the game is to pay as little of your paycheck as possible to a tax collector. Apparently kids love a good white-collar crime, because the game took off. It was distributed as an educational program for the Apple II and reproduced under many other titles for years. Although designed for kids, the game isn’t easy. Mathematicians have been publishing papers about it as recently as 2023 and still don’t know an optimal strategy. This week, you’re tasked with finding an optimal strategy for a specific small instance of The Taxman Game. File wisely and make your accountant proud.
Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!
You and a Tax Collector sit across a table with 12 paychecks on it. The paychecks are worth every whole dollar amount between 1 and 12 ($1, $2, $3, …, $12). You select paychecks for yourself one at a time, but every time you take a paycheck, the Tax Collector immediately takes all remaining checks whose values are factors of the number you chose. For example, if you choose the $8 check, then the Tax Collector will take the $1, $2, and $4 checks because 1, 2, and 4 are factors of 8. If the $2 check had already been claimed on a prior turn, then the Tax Collector would only take the $1 and $4 checks.
The Tax Collector must be able to take some paycheck on every turn. If you run out of legal moves (for example, if only paychecks $8 and $9 remain, then you can’t take either of them because their factors aren’t available to your opponent) then the Tax Collector takes all remaining paychecks. What is the largest amount of money you can claim for yourself in this game?
I’ll
be
back
next
Monday
with
the
solution
and
a
new
puzzle.
Do
you
know
a
cool
puzzle
that
you
think
should
be
featured
here?
Message
me
on
X
@JackPMurtagh
or
email
me
at
Did
you
go
all
in
on
last
week’s
poker
challenge?
Shout-out
to
Belenos
for
finding
the
natural
answer
and
accompanying
it
with
a
convincing
explanation.
Only
Eugenius
and
spudbean
got
the
bonus
puzzle
of
the
worst
starting
hand
that
still
forces
a
win.
Nice
work
all
around.
The most natural way for you to guarantee a win is to pick four 10s (the fifth card can be anything). A few other hands also work and I’ve listed them below, but they all involve taking at least three 10s. I found it a nice surprise that scooping up 10s was the key to winning in a poker puzzle, as opposed to shooting for a royal flush or four aces. I’ll lay out the argument with four 10s.
With no 10 available to me, the best hand that I can possibly end up with is a 9-high straight flush (because all straight flushes better than this contain a 10 and you have all of them). But no matter what I do, you can always end with either a royal flush or a 10-high straight flush, both of which beat my best possible hand.
For example, if I pick a 9-high straight flush, then you can discard four of your cards and make a royal flush with one of your 10s to clinch the win. On the other hand, if I pick four aces, then you can make a 10-high straight flush (or even king-high) with one of your 10s.
In general, if I do not prevent you from getting a royal flush, then you will get one and win. The only way I can prevent you from getting a royal flush is by taking a card away from every one of those hands (say, by taking four jack or four aces), but then you can make a different straight flush and still win.
By a similar argument, starting with the following hands also guarantees a win for you, provided that the non-10 cards do not share a suit with any of your 10s:
Three 10s with an ace and 9, king and 9, queen and 9, or jack and 9, Three 10s with a king and 8, queen and 8, or jack and 8, Three 10s with a queen and 7, or jack and 7, Three 10s with a jack and 6 (the worst).When you subscribe to the blog, we will send you an e-mail when there are new updates on the site so you wouldn't miss them.
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