Problem #60 HARD
The Sandstone Smuggler
Scenario Number Theory Logic Deduction
Problem Statement
Constable Preethi of the Jodhpur police has been tracking a smuggler who transports stolen sandstone sculptures hidden inside ordinary cargo trucks.
Her informant tells her: 'The smuggler uses a convoy of exactly three trucks. Each truck carries crates. The total number of crates across all three trucks is 24. Multiplied together, the number of crates per truck gives 192. Added together, they give 24.'
Preethi pulls out her notebook. 'So I need three numbers that add up to 24 and multiply to 192,' she says.
'There is one more thing,' the informant adds. 'The first truck always carries the most crates, and no two trucks carry the same number.'
Preethi nods and stares at her notebook. She needs to figure out the exact distribution so she can tell which truck at the checkpoint carries the most crates — and therefore, which truck to search first.
Three distinct positive integers sum to 24 and multiply to 192. The three numbers are in strictly decreasing order. What are they?
Answer & Quick Explanation
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The three trucks carry 16, 6, and 2 crates respectively. Verification: 16+6+2 = 24 ✓, 16×6×2 = 192 ✓, all distinct ✓. Constable Preethi should prioritise searching the truck carrying 16 crates.
Detailed Editorial Solution
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Set up the system: a+b+c = 24 and a×b×c = 192 with a > b > c > 0, all distinct integers. Factorise 192 first, then search through factor triplets for those summing to 24.
Step 1: Factorise: 192 = 2⁶ × 3 = 64 × 3. List its factor triplets (where each factor ≥ 1): (1,1,192), (1,2,96), (1,3,64), (1,4,48), (1,6,32), (1,8,24), (1,12,16), (2,2,48), (2,3,32), (2,4,24), (2,6,16), (2,8,12), (3,4,16), (3,8,8), (4,4,12), (4,6,8).
Step 2: Check sums: (1,1,192) → 194. (1,2,96) → 99. (1,3,64) → 68. (1,4,48) → 53. (1,6,32) → 39. (1,8,24) → 33. (1,12,16) → 29. (2,2,48) → 52. (2,3,32) → 37. (2,4,24) → 30. (2,6,16) → 24 ✓
Step 3: Triplet (2, 6, 16): 2+6+16 = 24 ✓. 2×6×16 = 192 ✓. All distinct ✓. Ordered: 16 > 6 > 2.
Step 4: Continue checking remaining: (2,8,12) → 22. (3,4,16) → 23. (3,8,8) → 19 (not distinct). (4,4,12) → 20 (not distinct). (4,6,8) → 18.
Step 5: Only (2, 6, 16) satisfies both constraints with all distinct values summing to 24.
Step 6: Constable Preethi should search the truck with 16 crates first — the largest load, most likely to contain the hidden sculptures.
Key Insight:
Factor triplet enumeration is the key technique. By listing all factor triplets of 192 and computing their sums, you avoid algebraic manipulation that can get messy. Uniqueness of the solution means the smuggler's distribution is unambiguous — a satisfying closure for the story.