*Welcome to this month’s edition of Problems to Ponder! Pose them in your classroom as a challenge, and try them out yourself. Have an interesting solution? Send it to thevariable@smts.ca for publication in a future issue of* The Variable*,* *our **monthly periodical.*

Practice need not be mindless. This month’s problems were chosen for their potential to engage students in the practice of a variety of basic skills while at the same time encouraging the mathematical practices of pattern-seeking, working systematically, generalizing, posing interesting questions, and more. Several of the problems have a very high ceiling!

Keep in mind that the particular numbers used in the problems can be changed to suit students’ skill levels.

Hundred-dollar Nim

This is a game for two players. Imagine that you have a pile of $100, and on your turn you can remove $1, $5, $10, or $25. Players alternate turns; the player to reduce the amount to 0 cents is the winner (and gets to keep the $100). What’s your strategy?

*Adapted from* Vennebush, P. (2011, July 11). 5 math strategy games to practice basic skills [Web log post]. Retrieved from https://mathjokes4mathyfolks.wordpress.com/2011/07/11/5-math-strategy-games-to-practice-basic-skills/

Biggest product

Pick a number: say, 25. Now break it up into as many pieces as you want: 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would also work. Now multiply all those pieces together. What’s the biggest product you can make? Pick another. What’s your strategy? Will it always work?

*Adapted from *Swan M., as cited in Meyer, D. (2013, April 16). [Confab] Tiny math games [Web log post]. Retrieved from http://blog.mrmeyer.com/2013/tiny-math-games/

Squares of differences

Draw a square, and pick four positive integers to go in each of the corners. For example:

Then, at the midpoint of each side, write the (positive) difference of the numbers at the two adjacent vertices:

Now connect the midpoints to form a rotated square inside the original square:

Repeat. What do you notice? Try with different sets of numbers; explore.

Will this always happen? What if we drew a triangle instead of a square?

*Adapted from* Squares of differences: Subtraction practice toward a greater purpose [Web log post]. (2011, April 27). Retrieved from http://mathforlove.com/2011/04/squares_of_differences/

BONUS: Twenty-nine

Find the most interesting property, not related to size, that the number 29 has and that 27 does not have.