Illuminations: Archimedes' Puzzle

Archimedes' Puzzle
The Stomachion is an ancient tangram-type puzzle. Believed by some to have been created by Archimedes, it consists of 14 pieces cut from a square. The pieces can be rearranged to form other interesting shapes. In this lesson, students learn about the history of the Stomachion, use the pieces to create other figures, learn about symmetry and transformations, and investigate the areas of the pieces.

Learning Objectives
 
By the end of this lesson, students will:
  • Investigate tangram-like puzzles with the Stomachion and arrange the pieces to create interesting shapes
  • Describe the symmetric properties of the puzzle pieces
  • Use correct vocabulary to describe rotations and reflections
  • Calculate the relative areas of the puzzle pieces (optional)

Materials
 
Archimedes' Puzzle Activity Sheet

Instructional Plan
The Stomachion is an ancient puzzle that is at least 2,200 years old. It consists of 14 pieces that can be cut from a 12 × 12 square, as shown below left. As with its cousin the tangram, the object of the Stomachion is to rearrange the pieces to form interesting shapes. Some of the many shapes that can be formed are shown below right.

 

 

It is not known whether Archimedes developed the Stomachion, though the puzzle was definitely known by the ancient Greeks. Because Archimedes wrote about the puzzle extensively, however, two of its alternative names are Loculus of Archimedes and Archimedes' Puzzle.

Prior to the lesson, copy the 14 pieces onto a transparency sheet and cut them out to use on an overhead projector.

You may wish to present some of the above history to students to begin this lesson. Explain that you will allow them to play with the Stomachion in just a few moments.

Explain that the Stomachion consists of 14 pieces, and display the pieces on the overhead projector. To get students thinking about symmetry, ask the following questions:

  • Are any of the pieces congruent to one another? How do you know? [Yes. There are two pairs of congruent triangles. One pair share the center point of the square in the figure above; these triangles have a base of 6 units and a height of 2 units. The other congruent triangles have a base of 6 units and a height of 4 units, and they appear in the upper right and lower left corners of the square. To show that the pieces are congruent, lay one over top of the other to prove that they are the same size and shape.]
  • Are any of the pieces similar to one another? How do you know? [Yes. The congruent triangles mentioned above are also similar, since congruence is a special type of similarity. To show that the pieces are similar, align the angles to show that they have the same measure.]
  • Do any of the pieces have rotational or reflexive symmetry? [No.]
  • What kinds of pieces appear in the Stomachion? [All of the pieces have 3, 4, or 5 sides; that is, they are triangles, quadrilaterals, and pentagons.]

After the warm-up discussion, distribute the Archimedes' Puzzle activity sheet. To begin, you may wish to distribute only the first two pages; the third page contains questions about the area of the pieces, which is an optional component for older or more advanced students.

Archimedes' Puzzle Activity Sheet Archimedes' Puzzle Activity Sheet

Allow students some time to cut out the pieces of the Stomachion. (To save class time, you can distribute the first page of the activity sheet the day before teaching this lesson and ask the students to cut out the pieces as homework.) Allow all students in the class to arrange the pieces to form the large right triangle shown at the bottom of page 1 of the activity sheet. Circulate among students to ensure that they are able to do so, offering assistance to those who need it. Then, tell students that you would like them to work in pairs to construct at least two of the shapes that appear on page 2. (Depending on time limitations, you can allow students to create many more shapes.) To ensure that students work together, specify that one student's set of pieces be used for one arrangement, the other student's set of pieces be used for the other arrangement, and no one is allowed to touch their partner's puzzle pieces.

After they have constructed two shapes, students should answer Questions 1–3 on the activity sheet. For students who finish quickly, allow them to construct more shapes after answering the questions. When all students have answered the questions, conduct a class discussion on Questions 2 and 3. Students should be able to identify the center of rotation for those shapes with rotational symmetry, and they should be able to identify the line of symmetry for those shapes with reflexive symmetry. In the figure below, the red shapes have reflexive symmetry; the green shapes have rotational symmetry; and the blue shape has both reflexive and rotational symmetry.

 

 

As a final part of the lesson, you may wish to have students compute the areas of the puzzle pieces. On page 3 of the Archimedes' Puzzle activity sheet, the 14 pieces are arranged in a 12 × 12 square configuration. When arranged as shown, all of the intersections occur on lattice points. Consequently, it is easy to calculate the area of each piece. Have students determine the area of each piece, and then discuss the results. (As a preliminary question for class discussion, you may wish to ask students to determine the area of the entire square.) In particular, students should notice that

  • The area of every piece is an integer.
  • More precisely, the area of every piece is a multiple of 3.

Questions for Students
 
How did you determine the area of each piece?
[For some pieces, the area can be computed directly using area formulas. For others, it may be necessary to surround the piece with a rectangle, compute the area of the rectangle, and then subtract the areas of the triangles that are not part of the piece.]

Assessment Options
 
  1. Ask students to create one of the shapes from the activity sheet and explain the various types of symmetry that are present. Students can write their responses in a journal. Assess the answers based on their use of the vocabulary surrounding rotations and reflections.

Extensions
 
  1. Have students search the Internet to find other shapes that can be created by rearranging the pieces of the Stomachion. Alternatively, have students arrange the pieces to create a shape of their own, and then draw the outline of the shape and give it to a classmate.
  2. Research the Stomachion on the Internet to learn about recent developments. In 1998, an ancient manuscript was discovered that contained much of Archimedes' work on the Stomachion puzzle. Allow students to report on what they learn from this article.
  3. Ask students to create a different arrangement of the 14 pieces (other than the arrangement shown on the activity sheet) to form a 12 × 12 square. Then, have them place their new arrangements on a grid. They should notice that the intersection points between pieces always occur at a lattice point. (This is interesting, because this property does not hold for most tangram-like puzzles.)
  4. Draw a tangram square on a 12 × 12 square grid. Do the intersection points always occur on lattice points?

Teacher Reflection
 
  • What material could you use to create more permanent Stomachion pieces?
  • How does the size of the puzzle pieces affect students' success in creating the various shapes?
  • In addition to having fun, were students able to learn and use the mathematical vocabulary and concepts associated with rotations, reflections, and symmetry?
  • What strategies could you use to improve classroom management for this lesson?
  • Were students able to work productively in pairs? Would it be more effective to have students work individually? in groups of four?

NCTM Standards and Expectations
 
Geometry 3-5
  1. Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.
  2. Identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
  3. Explore congruence and similarity.
  4. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.
Geometry 6-8
  1. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
  2. Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
Measurement 6-8
  1. Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.
This lesson prepared by Samuel E. Zordak.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics

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