Algebraic methods in plane geometry

In this article we examine the role of mappings in elementary geometry. After making some comments about the Erlangen programme initiated by Felix Klein in 1872, wherein he proposed a way of studying geometries based on the underlying transformation groups, we see how theorems like Von Aubel’s theorem and Napoleon’s theorem can be proved in an elegant manner using similarity mappings, and how some construction problems may be solved using isometries. At the end we present a recent proof by Alain Connes of “Morley’s Miracle”, based on affine transformations. Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject.     

Odd behaviour of the even integer 2

It is always of interest when one finds a property that is true only of some given object, or some given class of objects. For example, if n > 1 is an integer, then (n − 1)! + 1 is divisible by n just when n is a prime number. In this article we look at some properties that are true only for the integer 2. Shailesh Shirali has been at Rishi Valley School, Andhra Pradesh (Krishnamurti Foundation India) since the 1980’s. He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and on historical aspects of the subject. He has been involved in the Mathematics Olympiad movement at the national and international level for the past two decades. He is the author of several expository books and articles aimed at interested high school students.     

On sums of powers of integers

For non-negative integers k, n, let P _k (n) denote the sum {fx27-1}. We show by two different means that if k ≥ 3 and odd, then n ²(n+1)² iss a factor of the polynomial P _k (n); and if k ≥ 2 and even, then n (n+1) (2n+1) is a factor of the polynomial P _k (n). We also derive a relatively unknown result first obtained by Johann Faulhaber in the 17th century. Shailesh Shirali has been at Rishi Valley School, Andhra Pradesh (Krishnamurti Foundation India) since the 1980’s. He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and on historical aspects of the subject. He has been involved in the Mathematics Olympiad movement at the national and international level for the past two decades. He is the author of several expository books and articles aimed at interested high school students.     

The Sierpinski problem

In this article, we describe briefly a number-theoretic problem first studied by Sierpiński, now known as theSierpiński problem. The problem remains open.