🧩 THE IMPOSSIBLE BLOCK #03 — BULL'S SECRET — SOLUTION ❓ THE TWO HIDDEN NUMBERS: 4 and 13 Sum = 17 · Product = 52 🐂 BULL'S SECRET: Bull's sum is 17. And before anyone spoke, he knew every door of 17 was already LOCKED: every possible split gives a product that factors more than one way. Only 10 sums in the whole universe have that property. That hidden restriction is what made everything else possible. Rules: distinct integers from 2 to 99. Bull holds the SUM. Donkey holds the PRODUCT. Same logic as #02. 4,753 possible pairs. 🧩 1. WHY THE FIRST POSSIBILITIES DISAPPEAR 🫏 DONKEY: "I don't know." Any product with only ONE legal factorization is an OPEN door. Donkey would have known instantly, so those pairs die. 6 = 2×3 only · 8 = 2×4 only · 22 = 2×11 only · 77 = 7×11 only Every product of two different primes, and any product with a prime factor over 49 (like 53×4 = 212), is removed. 4,753 → 3,021 🧩 2. WHY BULL CAN MAKE HIS STATEMENT 🐂 BULL: "I knew you didn't know." Not maybe. Not probably. CERTAIN, from his sum alone. 🚪 THE DOORS Every split of Bull's sum is a door. OPEN = the product factors one way only (Donkey would know). LOCKED = the product factors more than one way. Bull can only be certain if EVERY door of his sum is LOCKED. Bull's doors for 17, all locked: 2+15 → 30 = 3×10 🔒 · 3+14 → 42 = 2×21 🔒 · 4+13 → 52 = 2×26 🔒 5+12 → 60 = 3×20 🔒 · 6+11 → 66 = 2×33 🔒 · 7+10 → 70 = 2×35 🔒 8+9 → 72 = 3×24 🔒 Sums with an OPEN door are eliminated: • Every even sum up to 54 → two different primes (8 = 3+5, 28 = 5+23, 54 = 7+47). Sum 6 too: 2+4 → 8. • Every odd sum where (sum − 2) is prime → 2×prime (13 = 2+11 → 22). • 51 → 17+34 → 578 = 17×34 only. • Every sum from 55 to 196 except 194 → a prime from 53 to 97 (57 = 53+4 → 212 only; 106 = 59+47). • 194 → 95+99 → 9405 only. 197 → 98+99 → 9702 only. ONLY TEN SUMS HAVE EVERY DOOR LOCKED: 11, 17, 23, 27, 29, 35, 37, 41, 47, 53 3,021 → 145 🧩