Figuring out a pattern for finding the square of, say, 10, then 11, then 12, etc. In my mind, the formula was: x^2 = (x - 1)^2 + (x - 1) + x.
For example, if you know that the square of 10 is 100, the square of 11 is 100 + 10 + 11 = 121. The square of 12 is 121 + 11 + 12 = 144.
Another way to think about it is enlarging a grid along both dimensions: first add an extra column (x - 1), then add another full row (x).
I "found" the pattern by observing and thinking about the relationships. I never came up with a formal proof - I'm a developer, not a mathematician! Here's a good explanation of why this works with intuitive images rather than just equations and words: https://betterexplained.com/articles/surprising-patterns-in-...
For example, if you know that the square of 10 is 100, the square of 11 is 100 + 10 + 11 = 121. The square of 12 is 121 + 11 + 12 = 144.
Another way to think about it is enlarging a grid along both dimensions: first add an extra column (x - 1), then add another full row (x).
I "found" the pattern by observing and thinking about the relationships. I never came up with a formal proof - I'm a developer, not a mathematician! Here's a good explanation of why this works with intuitive images rather than just equations and words: https://betterexplained.com/articles/surprising-patterns-in-...