Problem 18 · 2011 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintsubstitution
Nick wants to write whole numbers into the cells of the 3×3 table shown so that the sum of the numbers in each 2×2 sub-table is always 10. Five numbers have already been written. Determine the sum of the remaining four numbers.
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Answer: D — 12
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Hint 1 of 2
Each 2×2 block summing to 10 gives one equation among neighbouring cells.
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Hint 2 of 2
Add the right equations so the known centre and corners cancel, leaving the four unknown edge cells.
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Approach: combine the four 2x2 equations
- Label the grid with corners 1, 0, 4, 3 and centre 2; the unknowns are the four edge-midpoints b, d, f, h.
- Top blocks give b+d = 7 and b+f = 8; bottom blocks give d+h = 4 and f+h = 5.
- The needed sum is b + d + f + h = (b+d) + (f+h) = 7 + 5 = 12.
- So the four remaining numbers total 12, choice (D).
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