Roots and Precision

Computing $\sqrt{n}$ with floating-point math can give wrong answers. sqrt(49) might return $6.999999999$ instead of $7$. If you cast that to an integer, you get $6$.

For integer square root, use binary search instead. Search for the largest $k$ where $k^2 \leq n$. This avoids floating-point errors and runs in $O(\log n)$ time, $O(1)$ space.

Newton's method is another option: start with guess $x$, then set $x = (x + n/x) / 2$ using integer division. It converges fast, usually under $30$ iterations even for $64$-bit inputs.

When a problem asks "is $n$ a perfect square?" or "find the integer square root," avoid math.sqrt. Use binary search or Newton's method with integer arithmetic.