Problem 27 · 2020 Math Kangaroo
Stretch
Logic & Word Problems
sum-constraintcasework
Ana wants to write positive whole numbers, one in each of the squares shown, so that the sums of the four numbers in each row and the four numbers in each column are equal. She has already written some numbers, as shown. She wants to write the missing numbers so that the sum of these six numbers is as small as possible. What is this sum?
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Answer: B — 24
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Hint 1 of 2
All four row sums and all four column sums must equal one common value S.
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Hint 2 of 2
Each already-filled row or column caps how small its blanks can be, so push S down to the smallest value that still leaves every blank a positive whole number.
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Approach: minimise the common row/column sum, then add the forced blanks
- Every row and every column must add to the same total S, so the given numbers in any line bound S from below.
- Choosing the smallest S that keeps all six blanks positive whole numbers and fits every line, the blanks are forced.
- Adding those six forced entries gives the minimum sum 24, choice B.
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