by Dave Youngs
The Puzzle Corner activity this month is an adaptation of one of the more difficult matchstick puzzles from recreational mathematics. The challenge in the original puzzle is to make four equilateral triangles using six matchsticks. If you haven't seen this puzzle before, get out six toothpicks (flat ones work best) and try to solve it. This will give you a good idea of why the puzzle is one of the more difficult, yet ultimately rewarding, ones in recreational mathematics. It will also give you an appreciation of what your students will be asked to do in this activity. If you have seen the puzzle before, try to remember how difficult it was for you the first time you encountered it. (I remember being stumped by this puzzle for several days before making the leap of insight necessary to solve it.) In order to ease students into the classic puzzle described above, and to help alleviate a bit of the frustration they might encounter, I have added several warm-up challenges to my version of this puzzle.
The activity, Toothpick Triangle Challenges, includes four challenges, each of which asks students to build a different number of equilateral triangles using six flat toothpicks. The first two challenges are fairly easy and ask students to build one and two triangles, respectively. The third challenge, which is impossible, asks students to build three triangles. (While students may be able to make three triangles using the six toothpicks, the triangles will not be equilateral. Students with this alternate solution should be praised for coming up with three triangles, but reminded that equilateral triangles must have all three sides equal.) The fourth challenge is the same as the original brain teaser�make four equilateral triangles using the six toothpicks.
After working on the fourth challenge for awhile, students may think that it is impossible. Others may come up with a solution with four non-equilateral triangles (a square with two toothpicks crossed as the diagonals). At this point you may need to remind them that it is indeed possible to make four equilateral triangles and that this classic brainteaser has been challenging people for years.
Solving this last challenge will take persistence and thinking out of the box. People who are inveterate puzzle solvers will tell you that it is common to work on a problem like this for a long time with no success and then solve it quickly when coming back to it several days later. If students get too frustrated, thank them for working so hard and tell them they can put the puzzle away and work on it some other time. Remind them that mathematicians often do this when working on especially challenging problems.
When students solve this puzzle, ask them not to share the solution �doing so will rob others of the joy of discovery and the satisfaction of having solved the puzzle themselves. Instead of showing others how to do the puzzle, these students should be challenged to think of some hints that will help their classmates without giving away the solution. As the teacher, you can work with the students who have solved the puzzle to come up with a list of hints that can be posted in the classroom.