Problem 15 · 2020 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintarithmetic-sequence
Toninho wants to write strictly positive, consecutive whole numbers in the nine places of the figure, so that the sum of the three numbers in each diameter is equal to 24. What is the largest possible sum of all nine numbers?
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Answer: A — 81
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Hint 1 of 2
The centre number sits on every diameter, so it gets counted in all the pair-sums.
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Hint 2 of 2
Write the total of all nine numbers in terms of the centre value, then make the centre as small as possible.
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Approach: express the total via the shared centre and minimise it
- Each of the 4 diameters sums to 24 and shares the centre c, so the eight outer numbers total 4(24−c) = 96−4c.
- Total of all nine = (96−4c) + c = 96−3c.
- With nine consecutive integers the constraint forces the run 5..13 (centre 5), giving total 96−15 = 81.
- So the largest possible total is 81.
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