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222,222,222,222,222,222^2 -222,222,222,222,222,222,221^2 =?

A calculator or a computer cannot solve this problem. But we can.

We know 1^2-0^2 =1

2^2-1^2=3

3^2-2^2=5

4^2-3^2=7

There seems to be a pattern here. To find the difference between two adjacent squares, you just have to add the unsquared numbers together. We are not sure that this pattern goes on forever.

Let us draw some pictures.

each square is made from the previous one. To go from 2^2 to 3^2, add 2 dots to the bottom and 3 dots to the side. The difference between 3^2 and 2^2 is 3+2=5. It is clear from the pictures.

Similarly, to go from 3^2 to 4^2, add 3 dots to one side and 4 dots to another side. The difference here is 3+4=7. The picture shows the same. Now we understand that the pattern will go on forever.

This is proof by induction. In the same lines, we can get the answer to our problem. Just add the two numbers without squaring.

222 222 222 222 222 222 222 +222 222 222 222 222 222 221= 444 444 444 444 444 444 443

got it.

The essence of "proof by induction":

step1. Show it is true for the first one.

step2: show that if anyone is true then the next one is true.

Then all are true.

In other words: a Domino effect

Step1: The first domino falls.

step2: when any domino falls, the next domino falls

So all dominos will fall.

falls one by one

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