Problem 20 · 2017 Math Kangaroo
Medium
Algebra & Patterns
sum-constraintgrid
Emily wants to insert nine numbers into the 3 × 3 table so that the sum of the numbers in any two adjacent cells (cells with a common side) is always the same. She has already written two numbers into the table. How big is the sum of all nine numbers?
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Answer: D — 22
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Hint 1 of 2
Equal adjacent sums force a checkerboard: cells in the same colour class all hold the same value.
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Hint 2 of 2
The two given numbers belong to the two different colour classes.
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Approach: checkerboard pattern from equal adjacent sums
- If every pair of side-sharing cells has the same sum, the grid alternates two values like a checkerboard: 5 cells of one value, 4 of the other.
- The 2 sits in the 5-cell class and the 3 in the 4-cell class.
- Total = 5×2 + 4×3 = 10 + 12 = 22.
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