Problem 21 · 2022 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintcasework
Jenny writes numbers in a \(3 \times 3\) table so that the four numbers in every \(2 \times 2\) square have the same sum. Three corner cells are already filled in (see diagram). Which number does she write in the fourth corner?
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Answer: B — 1
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Hint 1 of 2
The four overlapping 2x2 squares all share the same sum; compare two of them that overlap in a row or column.
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Hint 2 of 2
Comparing the four equal block-sums forces a tidy rule on the four corner cells of the grid.
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Approach: equal block-sums link the four corners
- Comparing the four equal 2x2 sums cancels the shared middle cells and forces the two diagonal corner-pairs to have equal sums: one corner plus its opposite equals the other two corners added.
- The given corners are 2, 4 and 3; the missing corner pairs with 4 across the diagonal, so (missing) + 4 = 2 + 3 = 5.
- Therefore the fourth corner is 5 - 4 = 1, so the answer is B.
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