Problems to Ponder (December edition)

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Welcome to this month’s edition of Problems to Ponder! Have an interesting solution? Send it to for publication in a future issue of The Variable!

Primary Tasks (Kindergarten-Intermediate)

Colour-Coding Brownies [1]

Sam has brought a pan of brownies to a birthday party that has been cut into 24 equal pieces. He wants to share them equally among himself and his 5 friends at the party. Partition the pan of brownies and use colour coding to show how the brownies can be shared fairly. (Here is an example:

Here is another way in which Sam could share the brownies. Are there others?

Adaptations and extensions: What if there were 8 (or 12, or 9) kids at the party? What if the brownies had been divided into 30 (or 12, or 15) pieces?

Wooden Legs [2]

Wendy builds wooden dollhouse furniture. She uses the same kind of legs to make 3-legged stools and 4-legged tables. She has a supply of 31 legs. How many stools and tables might she make?

Extensions: How many stools and tables might she make if she must use all of the legs?

Frog Farming[3]

Farmer Mead would like to raise frogs. She wants to build a rectangular pen for them and has found 36 meters of fencing in her barn that she’d like to use.

a) Design at least four different rectangular pens that she could build. Each pen must use all 36 meters of fence. Give the length and with for each of the pens.

b) If each frog needs one square meter of area (1 m2), how many frogs will each of your four pens hold? Can you design a pen with the greatest frog capacity possible?

Intermediate and Secondary Tasks (Intermediate-Grade 12)

Mystery Box[4]

The areas of the faces of a rectangular box are 84 cm2, 70 cm2 and 30 cm2. What is the volume of the box?

Marbles in a Bag[5]

A bag contains 16 marbles, some black and the remainder white. Two marbles are drawn at random at the same time. It is equally likely that the two balls will be the same colour as different colours. How many balls of each colour are there inside the bag?

Extension: Suppose the bag contains x black balls and y white balls. What are the possible values of the positive integers x and y for which it is equally likely that two balls selected at random will be the same colour as different?

The Solid of Many Faces[6]

Draw a 3-dimensional picture of a solid shape that goes through each of the holes below, exactly touching every point on each of the sides as it passes through. (The object must be made of a solid that is not elastic and does not squash.)


[1] Adapted from Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages, and innovative teaching. San Francisco, CA: Jossey-Bass.

[2] Adapted from Ray, M. (2013). Powerful problem solving: Activities for sense making with the mathematical practices. Portsmouth, NH: Heinemann.

[3] Ray, M. (2013). Powerful problem solving: Activities for sense making with the mathematical practices. Portsmouth, NH: Heinemann.

[4] Bellos, A. (2016, December 5).  Can you solve it? Are you smarter than a Singaporean 10-year-old? The Guardian. Retrieved from

[5] Adapted from Barbeau, E. (1995).  After math: Puzzles and brainteasers. Toronto, ON: Walls & Emerson, Inc.

[6] Bellos, A. (2017, July 17).  Can you solve it? Are you smarter than an architect? The Guardian. Retrieved from

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