The Lighthouse of Vanishing Numbers

They say the Island of Aeloria appears only when the sea forgets its maps. At its heart sits a brass lighthouse whose keeper, an old clockmaker named Corin, carved five sealed lenses and a single code to open them. Each lens hides a single ray of light; only when all five rays align does the island name reveal itself.

“When hours fold like paper cranes, trace the small truths inside each vein. Solve the clock, collect the five — each a gate, each a clue, alive.”

Solve the five riddles below in order. Each answer is a single digit (0–9). Put the digits together A B C D E to form the lighthouse code. The correct code will open the lighthouse.

1. A — The First Lens (A):
   Corin’s first gear is engraved with the product of the first three distinct primes. Then he scratched out one digit from that product and wrote: “The number of distinct prime factors of the original product is your first lens.”
2. B — The Second Lens (B):
   On a brass disc he wrote a modular trick: “Count the residues r ∈ {0,1,…,p−1} that satisfy r² ≡ 1 (mod p), where p is the smallest odd prime greater than A. That count is your second lens.”
3. C — The Third Lens (C):
   Corin loved shapes. He said: “Find the smallest integer greater than 1 that can be written as a cube of an integer. That cube’s unit-digit is the third lens — but beware: if the unit-digit is 0 use 0.”
4. D — The Fourth Lens (D):
   He then described a drawing: “Draw a regular polygon with 8 vertices. Connect every second vertex to form a new polygon. How many sides does the resulting polygon have? That’s the fourth lens.”
5. E — The Fifth Lens (E):
   Finally, a tiny philosophical lock: “Consider the identity function on any finite set. How many elements remain fixed by every identity map? The smallest positive integer that matches that property is your fifth lens.”

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