4 hours ago
7
Earlier today I set you the following puzzle, suggested by Briton Alex Chui, the most successful ever candidate at the International Mathematical Olympiad.
Last month Alex, who is a pupil at Tonbridge School in Kent, became the first person to win a medal at the olympiad for seven years in a row. Bravo!
How many ways are there to fill a 3×3 grid with positive whole numbers such that the product of the numbers in each row and each column is 30?
[Each of the nine cells in the grid has a whole number in it. When you multiply the three numbers in each row, and in every column, the answer is 30. The three numbers in each row or column do not have to be three different numbers.]
Alex 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.”
Solution 216
The first thing to notice is that 30 = 2 x 3 x 5.
So, every row and column must include a 2, a 3 and a 5.
There are six ways to have a single 2 in every row and column. (3 ways to choose the 2 in first row, 2 ways to choose the 2 in second row in a different column, and 1 way to choose the last 2 in the third row, so 3 × 2 × 1 = 6). Here they are:
Similarly there are 6 ways to choose the placement of 3s and the placement of 5s.
Each full grid is a unique placement of 2s, 3s and 5s. Therefore there are 6 × 6 × 6 = 216 ways to fill the grid.
For example, here is one way to combine 2s, 3s and 5s.
These placements produce the full grid below, which is constructed by placing each number in the correct cell. If there is more than one number per cell you multiply them, and if there are no numbers, you put a 1.
There are 216 ways to combine the 2s, 3s and 5s in this way.
I hope you enjoyed today’s puzzle. Thanks to Alex and best of luck for in your mathematical career!
I’ll be back in two weeks.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
