The Tax Collector (aka The Tarot Card Riddle)

This is an updated 3-part version of our classic writeup of the Tax Collector. This has become a classic puzzle since our writeup for the New York Times Numberplay column, and is featured as a quintessential Building Thinking Classrooms task here and here.

This version comes from our new Math for Love Supplemental Curriculum for 6th Grade, though the problem is great for any grade from 4th and up. Days 1 and 2 introduce and expand the original problem. Day 3 features a TED-Ed riddle that offers a new spin on the original concept: The Tarot Card Riddle.

The writeup below is only a piece of the entire lesson. Download the PDF to see everything.

Materials and Prep

Index cards (or sticky notes) labeled $1 – $12, scratch paper, pencil.

How to Play

Start with a collection of paychecks (cards), 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. The tax collector must receive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over, and the tax collector gets all the remaining paychecks. The goal of the game is to beat the tax collector and get as much money as you can.

Launch

This is the first time that students will play Tax Collector. They will have additional opportunities to play in later lessons.

Display the 12 numbered cards and demonstrate how to play by involving your students and playing the role of the tax collector yourself. Here’s how it could play out…

Teacher: Here are 12 paychecks, each with their value written on them. This is your money and I you want to get as much as you can. Which would you like first?

Students: $12!

Teacher: OK. If you get $12, the tax collector gets $1, $2, $3, $4, and $6. Why did I take those numbers?

Students: Those are the numbers that are factors of (“go into”) 12.

Teacher: Oops. I forgot to tell you this important rule. When you take an paycheck the tax collector takes any remaining factors. So if you get $12, the tax collector gets $1, $2, $3, $4, and $6. Why did I take those numbers?

Students: 11.

Teacher: Here’s another rule. If you to take a paycheck, the tax collector must get some money too. Taking $11 isn’t a legal move because the tax collector can’t take any money (because the $1 paycheck is already gone). Is there another paycheck you could take instead?

Students: We can take $10.

Teacher: If you take $10, what does the tax collector get? Students: $5 (because the $1 and $2 were already taken).

Teacher: Right. Now if you look at what’s left, you have no more legal moves, so the game is over, and the Tax Collector gets the remaining paychecks ($7, $8, $9, and $11). Let’s find our totals. You got $22. And the tax collector got $56.

If you played this game again, do you think you could do better than $22?

Work

Students work in pairs or trios playing the game, alternating who is the taxpayer. When groups feel like they are satisfied with their approach, they can discuss their strategies and solutions with another group. Do they agree? What is the same/different about their strategies and/or solutions?

If the class (or individual groups) are satisfied that their solution is indeed the best possible, they can try to extend their reasoning by playing the game with more paychecks such as $1-14.

Tips for the Classroom

1. Playing the part of the tax collector is a fun way to launch and engage students in this activity.
2. Avoid the temptation to be the answer key for your students. Instead, invite them to discuss with each other that they’ve found the best score possible.
3. When appropriate, focus on using vocabulary such as factors, prime numbers, and composite numbers.