The solution to the puzzle set earlier today is 216. The puzzle, suggested by Alex Chui, a pupil at Tonbridge School in Kent, asked for the number of ways to fill a 3x3 grid with positive whole numbers such that the product of the numbers in each row and each column is 30.
Key Insight: Prime Factors
Alex Chui, the most successful candidate in the history of the International Mathematical Olympiad, said: “I like this puzzle because the key idea is to look at the prime factors of the numbers and realize that the arrangements of each prime factor (2, 3, 5) can be considered independently. This clean separation into different parts is one of my favourite aspects about maths.”
Since 30 = 2 × 3 × 5, every row and column must include a 2, a 3, and a 5. The placement of each prime factor can be considered separately.
Counting the Arrangements
There are six ways to place a single 2 in each row and column: 3 choices for the first row, 2 for the second, and 1 for the third, giving 3 × 2 × 1 = 6. Similarly, there are 6 ways for the 3s and 6 ways for the 5s.
Each full grid is a unique combination of placements for 2s, 3s, and 5s. Therefore, the total number of grids is 6 × 6 × 6 = 216. For example, one combination of placements yields a grid where numbers in the same cell are multiplied, and empty cells are filled with 1.
The puzzle was set as part of the Monday puzzle series, which has been running since 2015. Alex Chui recently became the first person to win a medal at the International Mathematical Olympiad for seven consecutive years.
