This week’s Puzzle Corner is an original toothpick puzzle which will challenge students’ problem-solving and critical-thinking skills while helping improve their geometric vocabulary. This puzzle, and others like it, also stretch students’ spatial-visualization and manipulation abilities. As you present puzzles of this nature to your class, you will find that your students possess a wide range of spatial abilities. Often, students who are strong spatially are not the top students in the class. Puzzles like this give them a chance to excel while some of the traditionally “good” students may struggle a bit more. As with any puzzle, persistence is the key, so encourage students to persevere until they find a solution.

Each student will need 14 flat toothpicks and a copy of the student sheets. (Flat toothpicks are preferable to round toothpicks because they will not roll.) There are two presented, a set of warm-up challenges and a set of main challenges. Depending on the age and spatial development of your students, you may wish to present only one of these.

The warm-up challenges (student sheet one) use only 12 toothpicks, and all involve removing two toothpicks to leave specified shapes. Most of these warm-up challenges have multiple solutions. The warm-ups are fairly simple and few in number, so it should only take students one class period to discover at least one solution for each challenge.

The main challenges (student sheet two) use 14 toothpicks, but the basic 12-toothpick arrangement is the same as in the warm-ups. The difference is that for each challenge, two toothpicks from the original arrangement must be moved, and two must be added to create the specified shapes. Again, most of the challenges have multiple solutions. Because there is a total of 12 challenges, some of which may prove quite difficult, the second student sheet should be worked on over a period of several days to give students time to solve each one. Having several days to work on the problem will also reduce the frustration students feel if they are not able to discover solutions immediately.

All of the challenges use geometric terms such as triangle, parallelogram, trapezoid, irregular, congruent, etc. Be sure to introduce any vocabulary or concepts your students may not be familiar with before beginning the activity.

Students should be encouraged to sketch each solution that they discover for a particular challenge. These solutions can be sketched on the back of the student pages or on sheets of scratch paper. At the end of the puzzle-solving time, students can compare the different solutions they were able to discover for the various challenges. You may even wish to use space on a wall or bulletin board to compile a master list of the different solutions discovered by the class.

This activity can easily be extended by modifying the rules in a variety of ways (move three toothpicks and add two, move one toothpick and add three, etc.), or by having students create their own puzzles using the rules given. Encourage students to create their own challenges and explore beyond the restrictions of the puzzle.

Arrange 12 toothpicks on your desk to form the shape shown below.

**Warm-Up Challenges**

- Remove two toothpicks to leave exactly four congruent triangles.
- Remove two toothpicks to leave three congruent triangles and two congruent parallelograms.
- Remove two toothpicks to leave two congruent triangles and two congruent parallelograms.
- Remove two toothpicks to leave exactly three congruent triangles and a parallelogram.

Make a sketch of each solution you discover. There may be more than one way to solve a challenge.

Take 14 toothpicks, and arrange 12 of them to form the shape shown below.

**Challenges**

Move two toothpicks in the arrangement and add the two extra toothpicks to make:

- Two congruent triangles, two congruent parallelograms, and an irregular pentagon.
- Two congruent triangles, two congruent parallelograms, and a trapezoid.
- Two congruent triangles, two congruent parallelograms, and two congruent hexagons.
- Three congruent triangles and two congruent trapezoids.
- Three congruent triangles and two non-congruent parallelograms.
- Three congruent triangles, two congruent hexagons, and a trapezoid.
- Four congruent triangles and an irregular hexagon.
- Four congruent triangles and two congruent parallelograms.
- Five congruent triangles and a parallelogram.
- Exactly six congruent triangles.
- Six triangles, not all congruent, and two congruent parallelograms.
- Seven triangles, not all congruent, and a trapezoid.

Sketch each solution you discover. There may be multiple solutions for several of the challenges.

**Solution**

Click the arrow below to view the solution.

In *Transforming Toothpicks by Twos*, students were challenged to take a hexagonal array of toothpicks and create various geometric shapes by removing or moving and adding toothpicks. Click here to download a solution for each challenge. Keep in mind that there may be several solutions possible for a given challenge. The dashed lines indicate the toothpicks that were moved/removed.