Problem 17 · 2025 Math Kangaroo
Hard
Logic & Word Problems
caseworksum-constraint
Hassan writes either the number 0 or the number 1 in each field of the table. The sum in each row, each column and each diagonal should be exactly 3. Hassan has entered 0 in one field and then fills out the table completely. What is the sum of the numbers in the fields with question marks?
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Answer: B — 2
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Hint 1 of 2
Each row and each column has 4 cells but must total 3 using only 0s and 1s — so what does that force about how many 0s are in each row and column?
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Hint 2 of 2
There is exactly one 0 in every row and exactly one 0 in every column; place the rest of the 0s so the two diagonals also each have a single 0.
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Approach: each row and column hides exactly one 0
- A row of four 0s-and-1s that adds to 3 must be three 1s and a single 0, so every row has exactly one 0 — and by the same reasoning every column has exactly one 0, and each diagonal must also hold just one 0.
- Starting from the given 0, the ‘one 0 per row, one per column, one per diagonal’ rule pins down where all four 0s go, so the whole grid is forced.
- Looking at the four question-mark cells, two of them turn out to be 1 and two turn out to be 0, so their sum is 1 + 1 + 0 + 0 = 2, giving the answer (B) 2.
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