Sums of the first n odd integers

Abstract

There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.

Developing new picture proofs that the sums of the first n odd integers are squares

“…The goals for the collection of proofs were (i) personal satisfaction, (ii) possible future use as teaching materials, and (iii) with potential as materials for research on students' conceptions of rigour and insight. The chosen Theorem 1 and an initial collection of proofs, with commentary, has been published as [9], and the research with undergraduate students is now also submitted. This collection of proofs was sent to the second author (FST), who has a professional interest in the nature of proof and the philosophy of mathematical arguments.…”

“…(The naming convention used here, e.g. "Pictorial III", is to retain compatibility with names used in [9] for future disambiguation. )…”

“…This proof then triggered the idea of considering other ways of proving the result diagrammatically, and the ensuing discussion included the suggestion that the 'Rearranging II' proof from [9] could be turned into a diagrammatic proof.…”

“…Many proofs rely on this observation. The induction step in a typical proof by induction, including the 'Induction' proofs of [9], makes use of this observation directly to move from case to . So does the following 'Telescoping' argument.…”

“…The Telescoping argument is self-contained and appears very specific to this situation. The proof entitled 'Backwards reasoning' in [9] makes use of the Fundamental Theorem of Finite Differences.…”