Toothpick Puzzlers, by Michael Naylor
All you need is a box of toothpicks to challenge your students at the end of the year
As the school year winds down, you'll want some fun activities, games and puzzles that kids view as play. Little do they know, your students will be practicing geometry, visualization and even number sense and algebraic reasoning.
Quick pick (Grades K-4)
Working in pairs, one student makes an arrangement of five toothpicks and shows it to his or her partner for three seconds only, whereupon the other student attempts to build a copy of the pattern. Announce "Begin!" and then "Stop!" after three seconds, then allow half a minute for students to build and discuss with their partners. Play for four or six rounds and discuss interesting patterns, shapes or geometry concepts students used during their reconstruction.
Basic Nim (Grades 3-8)
First demonstrate this game on the overhead projector. Twelve toothpicks are spread on the table between two players. Players take turns drawing one or two toothpicks – the player who takes the last toothpick wins.
Have your students play a few times and try to find a winning strategy. Students will notice some winning positions: Leaving their opponent with three toothpicks will guarantee a win. This strategy can be extended: Leaving six toothpicks for your opponent will guarantee you'll be able to leave them three toothpicks on your next turn, which will win you the game. Can your students extend this strategy further?
When students have discussed strategies and understand how to win, challenge them with one of these variations: What if the person who takes the last toothpick loses? What if a player takes one, two or three toothpicks on a turn?
Classic Nim (Grades 5-8)
The original version of Nim is quite challenging. Place 15 toothpicks into three rows: a row of three, a row of five and a row of seven. Players may take one, two or three toothpicks from one row only. The last player to take a toothpick wins.
Tricky toothpick triangles
In this activity, students try to make triangles from different numbers of toothpicks. It encourages geometric and algebraic reasoning and uncovers a lot of interesting math ideas. Show your class a triangle made from seven toothpicks:
Ask your class to find all the triangles they can make with three to 12 toothpicks, and sketch them on a piece of paper. Label each triangle with its name. Use toothpicks to make the sides of the triangle only – no toothpicks should be used in the interior.
Here are a few questions to guide your students and make sure they're finding some important ideas.
Why can't you make a triangle with four toothpicks, and why is only one triangle possible with eight? Why aren't some triangles (like a one-five-one) possible?
What are the relationships between the number of toothpicks and the types of triangles possible?
What else have you learned about triangles by doing this activity?
Toothpick puzzles
These puzzles build spatial sense and visualization. Once your students have solved a few variations, challenge them to write their own. Hang your students' creations on the wall and encourage the class to try one another's puzzles.
Have your students make this figure with 12 toothpicks:
Have your students make this figure with nine toothpicks:
• Remove four toothpicks to leave one square.
• Remove four toothpicks to leave two squares.
• Remove two toothpicks to leave two squares.
• Remove three toothpicks to leave one triangle.
• Remove three toothpicks to leave two triangles.
• Remove four toothpicks to leave two triangles.
• Remove two toothpicks to leave two triangles.
• Ask students to then write their own problem.
Toothpick toughie (Grades 4-8)
This puzzle is the perfect challenge for students who think they know all the answers. Make four congruent equilateral triangles with six toothpicks. You may not bend or break any toothpicks; the side length of each triangle must be one toothpick exactly.
Don't give away the answer – let them puzzle over it for days. The secret: Make one triangle on the table with three toothpicks, then with the other three triangles, make the vertex and edges to complete a triangular pyramid.
Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.
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