Problems to Ponder (October edition)


British Columbia Association of Mathematics Teachers

Welcome to the October edition of Problems to Ponder! This month’s problems have been curated by Michael Pruner, president of the British Columbia Association of Mathematics Teachers (BCAMT). The tasks are released on a weekly basis through the BCAMT listserv, and are also shared via Twitter (@BCAMT) and on the BCAMT website.

Have an interesting solution? Send it to for publication in a future issue of The Variable, our monthly periodical.


I am calling these problems ‘competency tasks’ because they seem to fit quite nicely with the curricular competencies in the British Columbia revised curriculum. They are non-content based, so that all students should be able to get started and investigate by drawing pictures, making guesses, or asking questions. When possible, extensions will be provided so that you can keep your students in flow during the activity. Although they may not fit under a specific topic for your course, the richness of the mathematics comes out when students explain their thinking or show creativity in their solution strategies.

Because I am currently using these tasks with my own classes (Grades 8-12) and it is the start of the year, I am sharing tasks that are my favorites for building a problem solving and collaborating culture with students; as such, some of these tasks may already be familiar to many of you.

I think it would be fun and more valuable for everyone if we shared our experiences with the tasks. Take pictures of students’ work and share how the tasks worked with your class through the BCAMT listserv [which currently connects nearly one thousand educators from across the province, country, and even the world! –Ed.] so that others may learn from your experiences.

I hope you and your class have fun with these tasks.

Michael Pruner

Intermediate and Secondary Tasks (Grades 4-12)

September 5, 2016

The Tax Collector
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. 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 is to beat the tax collector.


Turn 1: Take $8. The tax collector gets $1, $2 and $4.
Turn 2: Take $12. The tax collector gets $3 and $6 (the other factors have already been taken).
Turn 3: Take $10. The tax collector gets $5.
You have no more legal moves, so the game is over, and the tax collector gets $7, $9 and $11, the remaining paychecks.

Total scores:
You: $8 + $12 + $10 = $30.
Tax Collector: $1 + $2 + $3 + $4 + $5 + $6 + $7 + $9 + $11 = $48.

Extensions: What is the highest score you can achieve? What is the lowest score? What if you had 18 paychecks?

Source: Antonick, G. (2015, April 13). The tax collector. The New York Times. Retrieved from

September 11, 2016

The Game of 31
Players take turns picking any number from 1 through to 6. Each time a number is picked, it is added to the total score. The player who makes the total score add to 31 wins.

Extensions: What if 31 loses? What if you can choose from 2 through 6?

Replacing Coins
On a table, there are 1001 loonies lined up in a row. I come along and replace every second coin with a nickel. After this, I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. After all this, how much money is on the table?

Extensions: Why is the repeating pattern 12? Design a task that has a repeating pattern of 15. How many starting loonies are needed to make a total of $100?

September 18th, 2016

Filling Jugs
You have a 3 L jug and a 5 L jug and an unlimited supply of water. How can you measure exactly 4 L of water?

Extensions: What if not 4 L? What if not 3 L and 5 L? What if the jugs were 3 L and 6 L? or 3 L and 7 L?

Flip a Card, Toss a Card

Start with the cards ace, two, three, four, and five. Arrange the cards in such a way that they come out in increasing sequence when you deal the cards out like this:

  1. top card – place on table
  2. next card – place at bottom of deck
  3. repeat this process until all cards are on the table.

What is the pattern? What is your strategy?

Extensions: Add more cards to the deck. Can you do 13 cards? 52 cards? 104 cards?

Source: Liljedahl, P. (2015, March 18). 15243 [Video file]. Retrieved fom

September 25, 2016

Using 4’s
Write each number from 1 – 10 using exactly 4 fours and any mathematical sign or symbol.

Extensions: 1 – 20 or 1 – 100. Can you do this with 4 fives, or 4 threes, etc…?

The Frog Puzzle
Three green frogs are trying to change position with 3 orange frogs. Green frogs and orange frogs can only move forward onto an empty lily pad or leap frog over a single frog onto an empty lily pad.
How many moves are required to solve this puzzle?
picture2Extensions: What if there were 4 frogs on each side (or 5 frogs, or n frogs…)? What if the number of frogs on each side is not equal? How can you communicate the solution to a friend over the phone?

Source: The frog puzzle. (n.d.). Retrieved from

Primary Tasks (Grades K-3)

September 25, 2016

How Many Are Hiding?


  • 10 or more snap cubes / objects per player
  • a cup for each player

In this activity, each child has the same number of cubes and a cup. They take turns hiding some of their cubes in the cup and showing the leftovers. Other children work out the answer to the question “How many are hiding,” and say the full number combination.

Example: I have 10 cubes and I decide to hide 4 in my cup. My group can see that I only have 6 cubes. Students should be able to say that I’m hiding 4 cubes and that 6 and 4 make 10.

Source: How many are hiding? (n.d.). Retrieved from

MichaelPrunerMichael Pruner is the current president of the British Columbia Association of Mathematics Teachers (BCAMT) and a full-time mathematics teacher at Windsor Secondary School in North Vancouver. He teaches using the Thinking Classroom model where students work collaboratively on tasks to develop both their mathematical competencies and their understanding of the course content.

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