SSD Persistence: A Mathematical System for Student Investigation

Abstract

One of the most difficult pedagogical exercises in teaching college mathematics is finding ways of showing students how professional mathematicians really work. Many college-level texts written in the theorem-proof-corollary-remark style help convince students that mathematics is actually discovered in the same way. At least somewhere along the line, students should be exposed to, and preferably experience for themselves, the search for pattern, the tentative generalizing, the further search for confirming or counter-examples, and the hypothesizing of a result, which in actual fact comprises a large part of the professional mathematician's work. Ideally, they should then continue on and confirm the result by proving it, or seeing it proved, deductively. Various parts of the history of mathematics are well suited to illustrate the above process: See, for example, the brilliant discussion of the development and proof of Euler's theorem for polyhedra in Lakatos [1]. However, it is much more satisfying for the student to be able to go through the process himself. This paper describes an investigation that may be undertaken with college students for this purpose. The structure is interesting, yet the ultimate deductive proof is short and simple enough for most students to be comfortable with. Sloane [2] has multiplied the of a number together to obtain a second number, then multiplied the of this second to obtain a third, etc. He defines the (multiplicative) persistence of a number as the number of iterations of this process required to produce a single digit number from the original one. (This paper is concerned exclusively with positive, whole, base 10 numbers.) For example, 76 has a persistence of three because it requires three steps to reduce it to one digit: 76 -*42 -*8. Of course, changing multiplication to addition in the above produces the old operation of casting out nines. A similar but perhaps more interesting operation may be defined as follows: Form a new number by squaring and then adding the of the old number. We will call this the sum of square digits or SSD operator, and denote it by -*. Thus, for example, 145 12 + 42 + 52, or 42.