I mentioned that binomial expansion of (n+1)2 gives n2 +2n + 1, which makes this far easier to do in your head than multiplying and adding columns. Bu there is a more general solution. (n+i)2 gives n2 +2i + i2. Now any power of 10 is easy to figure by factoring: 40 = 4 x 10 x 4 x 10 = 16 x 100. And therefore 452 is 402 + 2x5x40 + 52 = 1600 + 400 + 25 = 2000+25 = 2025. This is really fast.
And even as the digit count grows: 2022 = 2002 + 4 x 200 + 22 = 40,000 + 800 + 4=n 40,804. If you can factor by 2, awesome, but if you are within ten of a power of ten, this is easier. I love figuring out techniques, which is why as a software engineer I always looked for ways to solve a requirement by building a tool which had an application to solve the problem.
You have a very confusing typo, unfortunately, in this post. You say that "(n+i)^2 gives n2 +2i + i2". It should be "(n+i)^2 gives n2 +2ni + i2". You have it right in the example using 40 and 5.
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