Saturday, March 23, 2002 

THANK
you for your tremendous response to Mind Games. Letters have been coming
in regularly, but I had not anticipated the mailbag to be so big. I
shall get back to each one of you individually, but you'll have to give
me one more week for that. It is a pleasure to know that some of you
have developed interest in mathematics after reading this column, which
is its intended goal.
Most of you want to know what book I refer to for coming up with a problem every week; so, the answer is that there is no such book. I remember having written in this column that the book of all answers is a myth from schooldays and contrary to the true nature of mathematics. One should read the works of masters and not their pupils (By masters, I mean greats like Newton, Ramanujan and Fibonacci). Solving the problem
from the previous week  of three numbers raised to a certain power each
 requires that you have no calculator or computer. Ravinder Mittal,
Rajeev Kumar Tak and Amandeep Jindal came up with the correct answer,
but they did not tell me how did they arrive at it. Here's how the
solution has been explained in an ancient text, from the time when there
was no computer: First, for the three numbers A, B, and C, let's define
the function floor(x), where x is a real number, such that floor(x) =
the integer part of x. Let y = floor(loga (x)) + 1 . As a general rule,
y will be the number of digits of x in base a. If we reverse this, we
can say that x is somewhere between a y and ay + 1. Another basic rule
is loga (bc) = cloga (b). If we don't use this rule, the calculation
cannot be handled using any standard scientific calculators, as these
can't handle calculation with numbers greater than 10,100. If we use
these two rules to A, B and C in base 10, it will show that A has 6,609
digits, B has 6,606 digits, and C has 6,603 digits in base 10.
Therefore, A is bigger than B which in turn is bigger than C. A is the
biggest, and C is the smallest. 
There are many ways to solving a problem, which each solver may choose according to his or her personality. We'll talk more, till then, keep writing in at The Tribune or adityarishi99@yahoo.co.in. — Aditya Rishi
