The 31 Mirrors of Almarra — A Mind-Blowing Math Puzzle
The Story
On the fog-kissed isle of Almarra stands a ring of thirty-one ancient mirrors — each one numbered and fixed to a brass wheel. Every night a sea-sail arrives carrying a single lantern whose sail is stitched with the next prime number. The lighthouse keeper, following an old ritual, turns the ring clockwise by that prime raised to the night-number, always counting steps modulo 31.
Night 1: 21. Night 2: 32. Night 3: 53…
The Puzzle
Let pt be the t‑th prime. Each night t rotate the ring by ptt positions (mod 31). Find the smallest positive integer N such that:
∑t=1N ptt ≡ 0 (mod 31)
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