Problem 19 · 2017 Math Kangaroo
Hard
Algebra & Patterns
Logic & Word Problems
substitution
Jenny wants to write numbers into the cells of a 3×3-table so that the sum of the numbers in each of the four 2×2-squares are equally big. As it is shown in the diagram, she has already inserted three numbers. What number does she have to write into the cell in the fourth corner?
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Answer: D — 0
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Hint 1 of 2
Two 2×2 squares side by side share a whole column, so comparing them cancels those shared cells.
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Hint 2 of 2
Compare the two top squares and the two bottom squares, then subtract.
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Approach: compare side-by-side 2x2 squares so the shared column cancels
- Call the unknown corner x (bottom-right); the known corners are 3 (top-left), 1 (top-right), 2 (bottom-left). Write the middle-left cell as d and the middle-right cell as f.
- The two top squares are equal, and they share the middle column, so the leftover columns match: 3 + d = 1 + f.
- The two bottom squares are equal the same way: 2 + d = x + f.
- Subtract the second from the first: 3 − 2 = 1 − x, so 1 = 1 − x and x = 0, uniquely, choice D.
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