The Rule of Five: Digital Roots of Sums of Three Consecutive Squares

This note describes a pattern found by the author: if three consecutive natural numbers end in a multiple of 3, the sum of their squares always has digital root 5. The pattern was first observed by hand, then tested by computer on every qualifying group up to 3,000,000, on the last 500,001 groups below 4 quadrillion, and on one million randomly chosen groups below 4 quadrillion. No counterexample was found. A short algebraic proof shows the rule holds for every natural number. The same method gives a companion result: when the multiple of 3 is the middle number, the digital root is always 2. The result follows from known facts about remainders modulo 9, and is presented here as an independent discovery with a complete proof.