Can You Solve This IMO Puzzle? World's Smartest Teen Challenge 2026

Can you solve the IMO puzzle that challenged even the world's smartest teen? British student Alex Chui, the most successful contestant in International Mathematical Olympiad history, shared a gem: how many ways to fill a 3×3 grid with positive whole numbers so each row and column multiplies to 30? This brain teaser tests your combinatorial thinking and pattern recognition.

Understanding the 3×3 Grid Puzzle

The puzzle asks for the number of distinct grids where every row and column product equals 30. Since 30 factors as 2 × 3 × 5, each cell must contain a divisor of 30, but numbers can repeat. The challenge is counting all valid arrangements without brute-force listing.

Your first instinct might be to enumerate grids, but there are over 200 solutions. A smarter approach involves prime factorization and combinatorial placement of factors across rows and columns.

Prime Factorization Strategy

Each cell is a positive integer; for the product to be 30, each cell's prime factors must come from {2, 3, 5}. Represent each cell as 2^a × 3^b × 5^c, where a, b, c are non-negative integers. Then, for each row and column, the sum of exponents for each prime must equal 1.

This transforms the puzzle into counting 3×3 matrices of exponent triples, where each row and column sums to (1,1,1) for each prime separately. Since primes are independent, you can count for one prime and cube the result (or combine via multiplication principle).

Data: Number of Solutions by Prime

For each prime, the number of 3×3 matrices with row and column sums equal to 1 is 6 (these are permutation matrices). Since the primes are independent, total grids = 6 × 6 × 6 = 216, but some grids may be identical due to factor combinations? Actually, each combination of three permutation matrices yields a unique grid because the exponent triples determine the number uniquely. Hence, 216 ways.

Key Takeaways for Puzzle Solvers

• Prime factorization simplifies complex product constraints.
• Break down the problem by independent primes.
• Use combinatorial counting instead of brute force.
• Always check for symmetry and repetition.
• Practice with smaller grids to build intuition.

Why This Puzzle Matters

This puzzle exemplifies the elegance of mathematical olympiad problems. It requires creative restructuring, not just calculation. For students and enthusiasts, solving such challenges sharpens logical reasoning and problem-solving skills.

Alex Chui's perfect score at the IMO highlights the pinnacle of teen mathematical achievement. His suggested puzzle is accessible yet deep, making it perfect for anyone wanting to test their own abilities.

Step-by-Step Solution Approach

First, realize each cell must be a divisor of 30: 1, 2, 3, 5, 6, 10, 15, or 30. But the exponent method is more systematic. For each prime, you need a 3×3 matrix of 0s and 1s (since sum per row/col is 1) – that's a permutation matrix. There are 3! = 6 such matrices.

Since the primes are independent, you choose one permutation matrix for 2, one for 3, and one for 5. Each combination gives a unique grid because each cell's value is 2^a × 3^b × 5^c. Thus total = 6^3 = 216.

But wait – does the order of primes matter? No, because the grid is determined by the triple of exponents. So 216 is the answer.

FAQ

Now that you know the solution, try solving similar puzzles to sharpen your mind. Share this challenge with friends and see who can match the world's smartest teen!