Problem 29 · 2025 Math Kangaroo
Stretch
Logic & Word Problems
sum-constraintcasework
Patricia has written a number in each box of a \(7 \times 10\) table. The sum of the numbers in each rectangle of size \(3 \times 4\) or \(4 \times 3\) is zero. Patricia reveals two of the numbers, as shown in the diagram. What is the sum of all the numbers in the table?
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Answer: D — -45
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Hint 1 of 3
Subtracting two overlapping zero-sum rectangles that differ by one row (or column) shows two far-apart cells must be equal.
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Hint 2 of 3
This forces the entries to repeat with a small period, so most of the \(7\times10\) grid can be tiled by zero-sum blocks.
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Hint 3 of 3
Whatever cells are left over after tiling are pinned to the two revealed numbers 20 and 25.
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Approach: tile with zero-sum blocks, then add the leftover cells
- Two \(3\times4\) blocks sharing three columns but shifted one column differ only in their end columns, so those columns have equal sums; the same holds vertically, forcing a repeating pattern.
- Most of the \(7\times10\) board splits into \(3\times4\)/\(4\times3\) rectangles that each sum to 0, leaving only a few cells whose values equal \(-20\) and \(-25\) by the periodicity.
- Adding those leftover cells gives total \(= -(20+25) = \textbf{-45}\), choice (D).
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