Problem 15 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Jilly makes up a multiplication magic square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100. The products of the numbers in each row, column and diagonal should be equal. In the diagram it can be seen how she has started. Which number goes into the cell with the question mark?
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Answer: B — 4
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Hint 1 of 3
Multiply all nine numbers to find the constant product, then take its cube root for the centre.
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Hint 2 of 3
Use the constant product along a row, column and the diagonal to fill the cells.
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Hint 3 of 3
The product of all nine values is 10^9, so each line multiplies to 1000.
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Approach: multiplicative magic square
- All nine numbers multiply to 109, so each line's product is the cube root, 1000, and the centre is 10.
- Top row 20×1×? = 1000 gives 50; the diagonal 20×10×? = 1000 gives 5 at the bottom-right.
- The right column 50×?×5 = 1000 forces the marked cell to be 4.
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