Reading the Constraints

Assessment problems are set by people who choose $n$ deliberately. The bound is not flavour text, it is the author telling you which complexity class they intend, and once you can read it the space of plausible approaches collapses from everything you know to two or three things.

The ladder

These are the bounds problem authors set, and they are calibrated for a compiled language at a few seconds:

Bound on $n$	Target	Usually means
$n \le 12$	$O(n!)$	Permutations, brute force over orderings
$n \le 25$	$O(2^n)$	Subsets, meet in the middle
$n \le 500$	$O(n^3)$	Floyd-Warshall, interval dynamic programming
$n \le 5000$	$O(n^2)$	Pairwise dynamic programming, nested scan
$n \le 10^5$	$O(n \log n)$	Sorting, heap, binary search, balanced tree
$n \le 10^6$	$O(n)$	One pass, hash map, two pointers, prefix sums
$n \le 10^{18}$	$O(\log n)$	Binary search, arithmetic, fast exponentiation

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