Problem 18 · 2021 Math Kangaroo
Stretch
Algebra & Patterns
substitutionsum-constraint
A 3×3 square initially has the number 0 in each of its cells. In one step, all four numbers in one 2×2 sub-square (such as the shaded one) are increased by 1. This operation is repeated several times to obtain the arrangement on the right. Some numbers in this arrangement are hidden. What number is in the square with the question mark?
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Answer: C — 16
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Hint 1 of 2
Each cell's value is how many of the chosen 2×2 squares cover it; the centre is covered by all four.
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Hint 2 of 2
Centre minus a touching edge gives the opposite-corner pair sum; use the known 13 to split it.
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Approach: express cells via the four 2x2 operation counts
- Let the four 2×2 squares be used a, b, c, d times; the centre cell equals a+b+c+d.
- The top-middle cell equals one pair (here 18) and the centre is 47, so the other pair sums to 47-18 = 29.
- That other pair is the two bottom corners; one corner is 13, so the marked one is 29-13.
- The question mark is 16, choice (C).
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