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WHAT IS A PROOF?

      We have over 300 different proofs for the Pythagoras theorem. Is proving easy? Can you prove anything?

    Let us take one example. Consider the following map.


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    How many colors, (minimum) you need to color this map. No adjacent state should have the same color. You can say with confidence "At least 4 different colors are required". Are 4 colors enough to color any map? 

    Is there any map where five colors will be needed. Nobody had the answer until 1976. But four crayons seemed to be sufficient for any map. Nobody felt the need for the fifth color. At the same time, no one was able to prove that only 4 colors are a minimum requirement with certainty.

    It is thrown as a challenge to the math community. They were forced to answer the question "what is a proof?

    After the computers came into being, In 1976 with the aid of hundreds of computer processing time, the mathematician Appel and Haken firmly turned the four-color conjuncture into a four-color theorem. And map makers who had known it for years said, "told you so".

    Back to "what is proof". Mathematical proof is 100 percent, concrete certainty. And the proof lies at the very heart of mathematics."

    There are several methods to find the proof. Let us learn one way here.

    A man has ten blue socks in his drawer. He wants to take out one correct pair of socks(same color). How many socks he has to take out (randomly) to get the correct pair?

    Suppose he takes out two socks.  It may be black, blue or blue, black or black and black or blue and blue. We cannot be certain. Suppose, he pulls 4 socks from the drawer, it may be blue, black, blue and black, and so on and on. He has to try all the 200000 different combinations before arriving at an answer. But, it will take a long time.

    But we can prove that "only 3 socks are needed to get a correct pair' using logic.

    First, he has to take out 2 socks. If they have the same color, he will be having the correct pair in his hands. If they have different colors, he has to take out one more sock. The third sock is going to be one of these two colors, the three socks must certainly contain a pair. Hence three socks is the maximum needed.

   Hene proving is not that easy. But once proved, it is certain forever.

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