Gizmodo Monday Puzzle: Tax Evasion for Kids

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.

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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!

Puzzle #38: Tax Evasion

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 gizmodopuzzle@gmail.com

Solution to Puzzle #37: Transparent Poker

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: