# Does the falling tree make a sound if no one if there to hear it? (or communication via encryption)

Elizabeth Gamburg (gamburg@dimacs.rutgers.edu)
Janet Taubin (taubin@dimacs.rutgers.edu)

General Information
Using games and cryptography to promote the learning and understanding of transformational geometry and modulo systems.

Category
Geometry, cryptography, and modular arithmetic

The Problem
How can we promote communication while ensuring privacy and security? This is a question dating back to Roman times, when the first known code was Ďcreated by Caesar (The Caesar Cipher) the science of encryption is necessary for more than just military and national security reasons. Today, with such rapid development of communication tools, computers, e mail and electronic cash, the Ďneed for corporate and personal security and privacy is overwhelming. Mathematics is the driving engine behind the science necessary for the maintence of this security and privacy and thus ensures the flow of information via public Ďcommunication channels.

These puzzles can be used for all levels of mathematics students, depending on its complexity, from elementary school to the upper high school grades.

Suggested Materials
Graph paper with large scale
Pencils.

Prerequisites
Simple knowledge of Cartesian Coordinates

Activity Description
Review with class the properties of translation function. Have students do puzzle,with teacher guidance of the first few steps. What do you think this is? What hints, if any, do you need to decode this? (Starting point, the translation) Given the coordinates of the starting point, given the key, the translation let them begin to decipher the message. This can be done with students working independently or as cooperative learning. What happens when the student reaches the bounds of the grid? What should they do? This leads to discussion of clock arithmetic, which is modular mathematics. Students should be led to understand that the dimensions of the grid, give them two different modular systems to work with (the horizontal and the vertical). Have students work on puzzles. Have students design their own coded messages using various transformation Ďfunctions. Devise discussion topics projects with the following, Reading, Research Writing in Math. Research & write the reasons for modern day cryptography. Have students design their own encryption, puzzles and games. Have students use the internet to find out about the leading topics of research in the field the cutting edge of new developments, theories and proofs. Check newspaper,s magazines, T.V. for current events pertaining to these topics. are the ethical concerns discussions going U.S. government policy. Are the fears based on reality or paranoia?

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