# Problems to Ponder (February edition)

Welcome to the February 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. This post features only a subset of the problems shared by Michael last month – head to the BCAMT website for the full set!

Have an interesting solution? Send it to thevariable@smts.ca 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 are 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.

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 so that others may learn from your experiences.

Michael Pruner

Palindromes
A palindrome is a number reading the same backward as forward. Consider a two-digit number: for example, 84. 84 is not a palindrome, so reverse the digits and add it to the original number: 84 + 48 = 132. This is still not a palindrome, so try it again: 132 + 231 = 363. 363 is a palindrome, so 84 can be called a depth 2 palindrome. Find the depth of all two-digit numbers.

Extensions: What about 3-digit numbers? What about the depth for the second time becoming a palindrome? What happens when you shade a Hundreds Chart according to the numbers’ depth?

Marching Band
Students in a marching band want to line up for their performance. The problem is that when they line up in 2’s, there is 1 student left over. When they line up in 3’s, there are 2 left students over. When they line up in 4’s, there are 3 students left over. When they line up in 5’s, there are 4 students left over. When they line up in 6’s, there are 5 students left over. When they line up in 7’s, there are no students left over. How many students are there?

Extensions: What if there are over 200 students in the band? What if there are 6 left over when lined up in 7’s?

Source: John Grant McLoughlin

Game of 22
Arrange four rows of cards from ace to four as shown below:

Two players alternately choose a card and add it to the common total. The winner is the player who makes 22 or who forces the other player to go beyond 22. What is a winning strategy?

Five Cubes
Using exactly five interlocking cubes, make as many shapes as you can so that all five cubes are touching the table. How many different shapes can you make?

Source: Spring 2011 problem set. (2011). Vector, 52(1), 59-60.

Subtraction Graph
Two play this game. Use two dice and markers.

Roll the dice and subtract the lower number from the higher. Put a marker in the first empty square above your answer. A player wins when they place a marker such that it is the first to reach the top of the graph.

Source: The Surrey School District

Box of Chocolates
Erica doesn’t like odd numbers, so the box of chocolates shown to the left meets with her approval. The problem is that she has to remove six chocolates from the box in such a way that she leaves an even number of chocolates in each row and each column. How might she do this?

Source: Kordemsky, B. A. (1992). The Moscow puzzles: 359 mathematical recreations. New York, NY: Dover Publications, Inc. (Original work published 1972)

Michael 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.