Problem 19 · 2012 Math Kangaroo
Hard
Logic & Word Problems
sum-constraintwork-backward
A number from 1 to 9 is to be written into each of the 12 fields of the table so that the sum of each column is the same. Also the sum of each row must be the same. A few numbers have already been written in. Which number should be written in the grey square?
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Answer: B — 4
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Hint 1 of 2
All three row sums are equal and all four column sums are equal; the whole grid holds the same total either way.
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Hint 2 of 2
Find one common total you can pin down from an almost-complete row or column, then chase the rest.
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Approach: find the common totals, then fill the forced entries
- The grid is 3 rows by 4 columns; given are row 1: 2, 4, _, 2; row 2: _, 3, 3, _; row 3: 6, _, 1, grey. Three equal row sums and four equal column sums share the same grand total, so 3·(row sum) = 4·(column sum).
- Column 2 is 4 + 3 + (row-3 entry); since every entry is 1–9, matching all column sums forces the common column sum to be 12 and the common row sum to be 16.
- Now the bottom row must total 16: 6 + (row-3 col-2) + 1 + grey = 16; column 2 = 12 makes the row-3 col-2 entry 5, leaving 6 + 5 + 1 + grey = 16.
- So the grey square is 4 (B).
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