Symmetry

"Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise" (Source: Wikipedia)

I have always been a great fan of symmetry. Not only it pleased my aesthetic sense but always made me curious to find out more and more symmetrical objects around me. I am sure symmetry would have in store answers to a lot of our unanswered questions. Certainly it can make life easy in CAT mathematics - dont get surprised - trust me it can !!!

Lot of times you would see a question like how many times a particular digit would come say in first n natural numbers. I mean I cant think of using symmetry more beautifully in mathematics than solving this problem.

For example let us say the question was how many times digit 6 would come in first 986 natural numbers. Most of you would be able to solve this problem with the following approach:

In first 100 numbers, 6 would appear in 10 places at Units place, 10 places at tens place, so every 100 there would be 20 times 6 so till 1000, there would be 20*10 sixes in units and tens places. Add another 100 for hundres place between 600 and 699. Hence 300 times till 1000 and subtract 1 six becuase of 996 not being counted in first 986 natural numbers, and the answer is 299. Answer is correct but it does take time.

Now, try solving this problem using symmetry. Assume writing all numbers from 0 to 999 in three digit format i.e. 000, 001, 002, 003, 004...... 996, 997, 998, 999. In this way there are 1000 numbers each with three digits. Hence total number of digits are 1000*3 = 3000. Now the symmetry comes into action. Each of the 10 digits from 0 to 9 must repeat equal number of times, hence digit 6 should come 3000/10 i.e. 300 times and subtract 1 six because of 996 not being counted!! 299!!!!!!!!! voila!! was not it quickand smart!! Symmetry (and Sonu!!) doing the magic.

I am sure no coaching institutes would teach this (untill they read my blog!!). But, again the idea I want to convey is to start inventing solutions to the problems. That is the best way you will remember it and may also help in boosting your confidence.