Problem 27 · 2017 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintcasework
Nine whole numbers were written into the cells of a 3 × 3 table. The sum of these nine numbers is 500. We know that the numbers in two adjacent cells (sharing a common side) differ by exactly 1. Which number is in the middle cell?
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Answer: D — 56
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Hint 1 of 2
Adjacent cells differ by 1, so the grid splits into two parity classes like a checkerboard around the centre.
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Hint 2 of 2
Express all nine entries in terms of the centre value and set the total equal to 500.
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Approach: write all cells relative to the centre, then use the sum
- Colour the grid like a checkerboard; neighbours differ by 1, so the centre and four corners share one parity while the four edge cells share the other.
- A valid tight filling is centre \(m\), each edge cell \(m-1\), and each corner \(m\) (every adjacent pair then differs by exactly 1).
- The total is \(m + 4(m-1) + 4m = 9m - 4\); setting \(9m - 4 = 500\) gives \(9m = 504\).
- So the middle cell is \(m = 56\), answer D.
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