Problem 30 · 2012 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Positive numbers were written in a 3 × 3 grid in such a way that the product of the numbers in each row and each column is exactly 1. The product of the four numbers in each 2 × 2 grid that can be found inside the 3 × 3 grid is 2. Which number is written in the centre of the 3 × 3 grid?
Show answer
Answer: A — 16
Show hints
Hint 1 of 3
Multiply all four 2×2 block products together — that is \(2^4 = 16\).
Still stuck? Show hint 2 →
Hint 2 of 3
Group the resulting factors by row and use that each row and each column multiplies to 1.
Still stuck? Show hint 3 →
Hint 3 of 3
Almost everything cancels, leaving just the centre value.
Show solution
Approach: multiply the four block products and cancel using the unit row/column products
- Label the grid a, b, c / d, e, f / g, h, i with every row product and every column product equal to 1, and each of the four 2×2 block products equal to 2.
- Multiplying the four block products gives \(2^4 = 16\) and, collecting factors, equals \(a b^2 c \cdot d^2 e^4 f^2 \cdot g h^2 i\).
- Group by rows: \(abc = 1\) and \(def = 1\) and \(ghi = 1\), so this reduces to \(b \cdot e^2 \cdot h\); the middle column \(beh = 1\), leaving just \(e\).
- Hence the centre value \(e = 16\) (A).
Mark:
· log in to save