Find Three Consecutive Integers That Sum to a Given Number

Let the three consecutive integers be x-1, x, and x+1. Their sum is 3x, so a solution exists exactly when target is divisible by three, assuming integers may be negative. The middle value is target divided by three, and the answer is [x-1, x, x+1]. Start by confirming that consecutive means difference exactly one and that integer values, rather than positive integers only, are allowed. If positive values are required, add the condition x-1 greater than zero. Avoid a search loop because the equation gives the answer directly. Check zero, a negative multiple of three, a nonmultiple, and numeric bounds when subtracting one from a minimum integer. This is O(1) time and O(1) space. Explain the derivation aloud; the signal is recognizing symmetry around the middle, not writing clever code.