Frances Chin (fchin@dimacs.rutgers.edu)

Frances Gasman (fgasman@dimacs.rutgers.edu)

**The Problem**

Attempts to send and receive coded messages date back 4000 years to Egyptian hieroglyphics. Throughout the years many different methods of sending and receiving secret messages have been developed. The 16th century Milanese mathematician Girolamo Cardano invented a geometric method which uses a device called a rotating grille to both encod and decode messages. Students will experience this method of coding by actual construction and use of the grille.

**Suggested Materials**

Graph paper - 1 cm. squares

Scissors

Pencils

Handouts

**Prerequisites**

Basic knowledge of rotations and reflections

**Guided Exploration**

*Part I
An exploration using a given message and a given grille*

The students are asked to find the hidden message

A | S | T | O | G | U |

F | T | 0 | X | H | E |

T | T | M | S | C | U |

T | R | U | H | V | T |

L | L | E | R | L | E |

D | H | E | Y | R | I |

No clues are given at first. Students work in pairs. Give only a short amount of time (~5 min.) for this attempt.

Give students each a rotating grille as shown below and again ask them to find the hidden message. These grilles should be prepared and cut (in the # marked squares) in advance. Hopefully, some if not all groups will be able to decode part or all of the message.

# | |||||

# | # | ||||

# | |||||

# | # | # | |||

# | # |

Follow this exploratory session with a brainstorming discussion on how the grille was used and how such a grille might be constructed. Pose appropriate questions similar to the following to help in the discussion.

1. Did the grille have to be in a certain position to work?

2. Was it necessary to rotate the grille once or more?

3. Does it matter in which direction you rotate?

4. Would it matter in which direction you rotate?

5. What ideas do you have for the construction of the grille?

*Part II
Geometric construction of the grille*

Design the rotating grille:

1. Mark off a 6 x 6 square for the grille on the graph paper.

2. Shade a small triangle in the upper left corner of the grille. This is important for later orientation.

3. Subdivide the grille into four quadrants A, B, C, D.

A | B |

C | D |

a. Number the squares in quadrant A from 1 to 9.

b. Rotate the grille 90 degrees counterclockwise and number the squares in the next quadrant from 1 to 9.

c. Repeat the directions in 2 twice more until the grille is completely numbered.

4. Starting in any quadrant, select and circle exactly one of the squares marked 1. Select and circle exactly one each of the numbers 2 through 9 from the numbers in the grille. Try to scatter your selections by rotating the grille as you selec. Check to be sure you have circled each of the numbers from 1 to 9 exactly once.

5. Cut out the circled squares.

The rotating grille is now ready for use in coding and decoding messages.

*Part III
Using the grille*

To write the message in coded form:

Write a 36 letter message on paper. Do not count spaces or punctuation as part of the 36 letters.

Place the grille on top of a 6 x 6 square of graph paper with the marked triangle at the upper left corner.

Code the message:

1. Write the first nine letters of the message one at a time through the holes of the grille as you would read them.

2. Rotate the grille 90 degrees counter clockwise and encode the next nine letters.

3. Repeat the directions in #2 to more times until the message is completely encoded.

Give a copy of the code and the grille to your partner.

To decode or read the message, use the grille just as it was used to code the message.

**Concluding the Exploration**

The guided exploration will most likely take a full class period. Students should be given sufficient time to code and decode at least two or three different messages using each others' rotating grilles.

As students prepare and use the grilles in coding and decoding messgaes several types of mathematical concepts may be addressed. Transformational geometry questions may be posed based on the 4 quadrant numbering of the grille. Matrix nomenclature and symmetries can be explored. Construction limitations of the grille as well as the number and types of possible grilles also provide for rich problem solving exercises. Finally, some of the vocabulary of cryptography can be introduced.

**Transformational properties of the grille**

Before numbering, the grille was divided into 4 quadrants.

A | B |

C | D |

The small 3 x 3 square in quadrant A was numbered 1 to 9.

1. Give the direction and amount of rotation needed to get from the number 3 in quadrant A to 3 in quadrant D.

2. Is there more than one answer to the question in A?

3. What rotation will take us from the 2 in quadrant B to the 2 in quadrant D? What is the direction of the rotation?

4. Which numbers are transformed by reflection? Explain.

**Matrix nomenclature and symmetries withing the grille.**

Consider the grille as a 6 x 6 matrix numbered in standard matrix notation. The four ones in the numbered grille are in the (1,1), (1,6), (6,1), (6,6) positions. This is a symmetry about the center of the grille.

1. Write the matrix positions of the four 2's in the grille and describe the symmetry that occurs.

2. Do the same for each of the four 3's, 4's, ..., 9's

**Construction limitations, numbers, and types of grilles**

1. What limitations are there in the construction of the grille? What care must be taken in cutting? Remember the grille must hold together.

2. How many possible 6 x 6 grilles are there?

3. Can the same kind of coding system be used with a 4 x 4 grille? a 5 x 5 grille? an 8 x 8 grille?

4. Are grilles of different shapes also possible

5. Suppose two different grilles are constructed and the coding is done as follows:

grille one is used to encode the original message m1 to a new message m2, then grille 2 is used to code m2 to a final message m3.

Describe how the message would be decoded.

Does the order in which the two grilles are used matter?

**Some introductory cryptography terms**

After working with the grille the students are ripe for learning some of the formal and informal jargon of cryptography.

Suppose Bob and Alice are the message sender and receiver respectively. Bob's message is called the **plain text**. Alice receives the **cipher text** which is the coded message. Alice's cipher text is **public** information. An eavesdropper, Eve, could intercept the message. Even if that happens, Eve would have some trouble decoding the message due to the fact that the **private key** which is the grille (owned by both Bob and Alice) is not available to her.

**References**

1. Kahn, David, The Codebreakers, Scribner, New York, 1996, pp. 143-145, 308-309.

2. Gaines, Helen Fouche, Cryptanalysis, Dover, 1939, pp. 26-29.

3. Reeds, Jim, "Cryptanalysis as Puzzles", DREI 97 Conference, Rutgers University, August 1997.