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Rounding up to Nearest Square

Alexander Ross • • 2 min read
Rounding up to Nearest Square
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Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner?

ie.

$2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. What it does when x = 4 or 9 doesn't really matter.

I could write a function to do this easily enough. ie.

while (sqrt(x) is not int) {
  x++;
}

But I'd rather not do a brute force approach. Any ideas?

Thanks!

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2 Answers

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You can take the ceiling of $\sqrt{x}$ and square it.

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I don't know if this is quite what you're looking for, but you can use the fact that $ \ (n+1)^2 \ = \ n^2 \ + \ (2n + 1) \ $ . Once you reach a perfect square $ \ n^2 \ $ , the next $ \ 2n \ $ integers would be the ones that get "rounded-up". This is at least reduces the process to a test for perfect-square and a counting loop.

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Alexander Ross
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Alexander Ross

Alexander Ross has covered the video game industry for a decade, writing deep dives on game design, esports tournaments, VR developments, and gaming culture.