Problems to Ponder (September edition)

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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 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

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

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:
difference 2Now connect the midpoints to form a rotated square inside the original square:

difference 3
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

BONUS: Twenty-nine

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

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