The Challenge of Replacing Letters with Numbers: What contents and strategies can be developed?

Abstract

In this paper we present a kind of mathematical challenge called cryptarithmetic, which consists of the replacement of letters with numbers. We consider such challenges as problems, because there are no ready formulas to solve them, making it necessary to use problem solving strategies. We seek theoretical support from the writings of authors like Polya, Schoenfeld and Guzmán who discuss such strategies, listing some of their suggestions. We also give a few examples of solutions, detailing the reasoning employed, and suggest an activity for middle school students based on a simpler case. The activity suggested is designed for a group of teachers and students, in such a way that each participant can think about replacing letters with numbers, look at them critically and be challenged, so that he/she may experience problem solving as something more interesting than working on routine exercises. We conclude with some comments on the need to introduce new topics, like cryptarithmetic, to teacher education courses, so that future teachers may become more capable of inventing or adapting activities that would help students to develop creativity. Keywords: Mathematical Challenges. Problem Solving Strategies. Cryptarithmetic.