Learning Mathematics - Enjoy Pure Mathematics

  Problem Solving - an Illustration M. Velrajan How to solve problems is illustrated with a simple, fascinating number game. Let S be a string of digits 0, 1, 2, . . . , 9 of length n and 1 ≤ m < n.  Replace the m th and n th digits of S by m th digit + 1 and n th digit + 1,  respectively, subject to 9 + 1 = 0  i.e. the addition is addition modulo 10 and interchange the first m and the remaining n - m digits to get the new string of length n. For example, suppose S = 03421678 and m = 6. Then  S      034216 78 S1    79 0342 17 Repeat the above for the new string of digits. The problem is whether we get the same string S after a finite stage and if so, after which stage. Think how to solve this problem . . . . . . . . . Understand . . . . . . . . Yes, the operation given first increases only the m th and n th place digits of S by 1, of course 9 + 1 = 0 then interchanges the first m and the remaining n - m digits.  Analyse Thi...

  Finite Groups   M. Velrajan As a relation between number theory and  group theory, usually we get the Euler theorem  and Fermat theorem from Lagrange theorem  on finite groups.  Now we explore some more connections between numbers and finite groups.   First we recall some known examples and results of groups. For any natural number n ≥ 2,  (ℤ n = {0, 1, 2, . . . , n-1}, ⊕ n ) is a cyclic group of order n. For any natural number n ≥ 3, the symmetric group of degree n,  S n is a non-abelian group of order n!.    The group of symmetries of  squares  D 4 = { R 0 = e, R 90 , R 180 , R 270 , H, V, D, D′}  is a  group of order 8. D 4 has 4 rotational symmetries of rotation  by multiples of π/2 and 4 reflection symmetries,  reflection about the lines joining the midpoints  of opposite sides and the lines joining the opposite vertices. D 4 = {a 0 = e, a, a 2 , a 3 , b, ab, a 2 b, a 3 b},  where...