Category Archives: Toothpick Puzzles

Transforming Toothpicks by Two

Transforming Toothpicks by Two

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.

transform2

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.

transform3

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.

Toothpick Puzzlers

Toothpick Puzzlers

This week’s activity consists of five related puzzles that should challenge students. Matchstick (I have substituted toothpicks) puzzles have been a staple feature of recreational mathematics for years. The puzzles presented here were adapted from some that appeared in The Moscow Puzzles by Boris Kordemsky (available from Dover Publications, Inc.)

This type of puzzle often requires patience and persistence to solve. These puzzles are easier for students who have well-developed spatial-relationship skills. Often, these students are not the “top” students. In fact, this type of puzzle is often very frustrating for those students who do well at traditional school tasks. Because of this, these puzzles are useful, challenging those who have an easy time with most school work and giving success to others who need it.

It is important to try these puzzles yourself. If you find them difficult, don’t despair; they are difficult for many adults, especially those who are linear thinkers. Don’t assume, however, that all of your students will have as hard a time with the puzzles as you did. You may be surprised (at how well some of them do at these puzzles).

A positive aspect of these puzzles, for both students and adults alike, is that they exercise spatial-relationship skills as well as logic skills. After solving one of the five puzzles, you learn to “see” things, both visually and logically, that help solve the remaining puzzles.

Have fun and good luck!

Puzzle 1: Make two squares of different sizes by removing two toothpicks.

Puzzle 2: Make three congruent squares by moving three toothpicks.

Puzzle 3: Make three congruent squares by moving four toothpicks.

Puzzle 4: Make seven squares, not all congruent, by moving two toothpicks. You may cross one toothpick over another.

Puzzle 5: Make 10 squares, not all congruent, by moving four toothpicks. You may cross one toothpick over another.

Toothpick Puzzlers

Solutions

Click the arrow below to view the solutions.

The dashed lines in the diagrams indicate the toothpicks that were moved, the dotted lines indicate toothpicks that were removed.

Puzzle 1: Make two squares of different sizes by removing two toothpicks.

toothpick puzzlers sol 1

Puzzle 2: Make three congruent squares by moving three toothpicks.

toothpick puzzlers sol 2

Puzzle 3: Make three congruent squares by moving four toothpicks.

toothpick puzzlers sol 3

Puzzle 4: Make seven squares, not all congruent, by moving two toothpicks. You may cross one toothpick over another.

toothpick puzzlers sol 4

Puzzle 5: Make 10 squares, not all congruent, by moving four toothpicks. You may cross one toothpick over another.

toothpick puzzlers sol 5

If you’re interested in more fun puzzles check out the Puzzle Play book for Grades 4-8.

The 36 ‘Picks Puzzle

The 36 ‘Picks Puzzle

This puzzle comes from a rich historical tradition that dates back to the 19th century when matches were first manufactured. Invented in 1827 by the British chemist John Walker, matches soon replaced the tinder boxes and flints that people had formerly used to light fires. As matches grew in popularity and became ubiquitous later in the century, they spawned a new form of entertainment—matchstick puzzles—which became quite popular when several match companies printed these puzzles on their boxes. Capitalizing on this interest, publishers began to print books of matchstick puzzles. Near the end of the 19th century, many people had developed a personal repertoire of these puzzles that they used to challenge friends and acquaintances. The toothpick puzzle presented here is modeled after the classical matchstick puzzles. For safety reasons, these puzzles use flat toothpicks instead of matches. (Round tooth-picks are not recommended, as they tend to roll.)

This puzzle has six challenges, each of which starts with 36 toothpicks arranged to form 13 small squares. Students then either move or remove a given number of toothpicks to form the numbers of geometric shapes stated. Once students have a solution, they should use the dot paper to record it. The dot paper can also be used to solve the challenges if toothpicks aren’t available—students can simply draw the figure and then erase lines instead of removing or moving toothpicks. Several of the challenges have multiple correct answers. Students can be encouraged to find all of the possible answers for each challenge.

The challenges presented here require patience and persistence to solve. However, they tend to be a bit easier for students who have well-developed spatial-relationship skills. Often, these students are not the top students, and their ability to solve these puzzles faster than their peers is a great esteem builder. Conversely, this type of puzzle often frustrates those students who usually do well at traditional school tasks and provides them with a real challenge. This role reversal is often beneficial for both sets of students.

In addition to using their spatial skills, students can also utilize logical thinking when working on these puzzles. While each challenge may eventually be solved by trial-and-error, taking a few minutes to think logically about the problem will often reveal the solution. Another key puzzle-solving trait that students will need to develop when working on these puzzles is persistence—students can’t solve a puzzle if they give up. You will need to encourage students to be persistent and to keep trying until they solve the puzzles.

As the teacher, you are encouraged to try the puzzles yourself before giving them to your students. You may find them difficult, as do many adults who are linear thinkers, but don’t assume that your students will experience this same level of difficulty. You may be surprised at how well some of them do with these puzzles. Good luck!

Arrange 36 flat toothpicks to form 13 small squares as shown below. Use the toothpicks to solve the challenges.

36PicksPuzzle

36PicksGuyChallenges:

  1. Remove 4 toothpicks to leave 8 small squares.
  2. Move 6 toothpicks to make 14 small squares.
  3. Remove 8 toothpicks to leave 9 small squares.
  4. Remove 4 toothpicks to leave 5 small squares and 4 rectangles.
  5. Move 4 toothpicks to make 1 rectangle and 10 small squares.
  6. Remove 8 toothpicks to leave 5 small squares and 1 large square.

Click here to download and use the dot paper to record your puzzle solutions.

The Three-to-Five Triangle Puzzle

The Three-to-Five Triangle Puzzle

This week’s Puzzle Corner activity comes out of a rich historical tradition that dates back to the 19th century when matches were first manufactured. Invented in 1827 by the British chemist John Walker, matches soon replaced the tinderboxes that people had formerly used to light fires. As matches grew in popularity and became ubiquitous later in the 19th century, they spawned a new form of entertainment—matchstick puzzles—that became quite popular when several match companies printed these puzzles on their boxes. Capitalizing on this interest, publishers began to print books of match-stick puzzles. By the turn of the 20th century, many people had developed a personal repertoire of these puzzles and used them to challenge friends and acquaintances. The toothpick puzzle presented here is modeled after these classical matchstick puzzles, but for safety reasons it uses flat toothpicks instead of matches.

This puzzle may require patience and persistence to solve. It will be a bit easier for any students who have well-developed spatial-relationship skills. Often, these students are not the top students and their ability to solve puzzles like this one faster than their peers is a great esteem builder. Conversely, this type of puzzle often frustrates those students who usually do well at traditional school tasks and provides them with a real challenge. This role reversal has the potential to be beneficial for both sets of students.

Your challenge in this puzzle is to move exactly 3 toothpicks in the following arrangement to make 5 triangles. Good luck!

Triangles

Solution

Click the arrow below to view the solution.

The challenge was to move only three of those toothpicks to create a total of five triangles. This can be accomplished by moving the triangle on either the left or right above the other two triangles. This gives you four small triangles and a fifth large triangle.

Triangles Solution

Reducing Squares

Reducing Squares

Reducing Squares belongs to a category of puzzles called “matchstick puzzles” which were very popular in America during the last century. Most adults in those days carried small boxes of matches with them to light the many candles or lamps in their homes. Many of these same people had a favorite repertoire of matchstick puzzles to share with friends. Matchstick puzzles were just one of the indicators of a strong general interest in puzzles of all kinds during the 1800s newspapers in those days included a much greater variety of puzzles than today’s ubiquitous crossword puzzles and word jumbles.

In Reducing Squares toothpicks are substituted for matchsticks for safety reasons. I have found that flat toothpicks work the best because they do not roll. Each student will need 24 toothpicks for this particular puzzle. You may wish to have students make small paper envelopes in which to keep their toothpicks when they are not working on the puzzle.

Arrange 24 toothpicks as shown below. Each challenge asks to you reduce the number of squares in the arrangement by taking away different numbers of toothpicks. See if you can solve each challenge, and then come up with a challenge of your own. Make a record of your solutions.

Toothpicks

Challenge #1: Reduce the number of squares to two by removing eight toothpicks.

Challenge #2: Reduce the number of squares to five by removing four toothpicks.

Challenge #3: Reduce the number of squares to five by removing eight toothpicks.

Challenge #4: Reduce the number of squares to nine by removing four toothpicks.

Challenge #5: Create you own challenge and make a record of it.

Solutions

Click the arrow below to view the solutions.

Reducing Squares asked students to remove various numbers of toothpicks from an arrangement to leave different numbers of squares. The solutions appear below. Please note that in some situations, the solution shown is only one of several possibilities.

Reducing Squares Solutions

Flipping Fish

Flipping Fish

This puzzle has been around in various forms for a number of years. All forms begin with eight toothpicks or matches arranged in the shape of a fish. One version challenges you to move exactly three toothpicks to make the fish face the opposite direction. Another version challenges you to move just two toothpicks to make the fish face a different direction. Other versions add a small object for an eye that is moved along with the toothpicks.

I have slightly modified this classic puzzle by making it more open-ended and by asking students to think about their solutions. Flipping Fish begins by simply asking students to make a toothpick fish face another direction by moving any number of toothpicks. With this approach there are many possible solutions, instead of the one or two possible in the classic versions. When students discover an initial solution, they are asked to examine that solution and determine if they can come up with a better one (one that moves fewer toothpicks). They are then challenged to find the minimum number of moves in order to get the fish to face a different direction, and to defend this number. This strategy forces students to think more deeply about the problem and helps develop higher-order thinking skills.

We hope that you will enjoy using Flipping Fish with your students!

Arrange eight toothpicks in the shape of a fish. Move toothpicks until the fish is facing another direction. How many did you move?

stickfish

Try to get the fish to change directions by moving fewer toothpicks than you did the first time.

What is the minimum number of toothpicks that must be moved to make the fish face another direction?

Defend your answer.

Solutions

Click the arrow below to view the solutions.

In this puzzle Flipping Fish, students were challenged to make a toothpick fish face in another direction by moving toothpicks. There are many possible solutions to this puzzle; two are given below.

fish