Problem 28 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintcasework
Susi writes a different positive whole number on each of the 14 cubes of the pyramid (see diagram). The sum of the numbers on the nine cubes on the bottom is 50. The number on every other cube equals the sum of the numbers on the four cubes directly underneath it. What is the biggest number that can be written on the topmost cube?
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Answer: E — 118
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Hint 1 of 3
Each cube above is the sum of the four directly under it; the top is a weighted sum of the nine bottom numbers.
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Hint 2 of 3
The centre bottom cube counts 4 times, edge-centres twice, corners once; the nine distinct numbers total 50.
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Hint 3 of 3
Put the largest values where the weight is biggest to maximise the top.
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Approach: weighted sum of the bottom layer
- Adding up the pyramid, the top equals a weighted sum of the nine bottom numbers with weights: centre 4, the four edge-middles 2 each, the four corners 1 each, so top = (sum of all nine) + 3·(centre) + (sum of the four edges) = 50 + 3·centre + (edge sum).
- To make this biggest, push value into the centre: give the four corners the smallest distinct values 1, 2, 3, 4 (sum 10), leaving 40 for the centre plus four edges.
- Make the four edges the next-smallest distinct values 5, 6, 7, 8 (sum 26), so the centre is 40 − 26 = 14; then top = 50 + 3·14 + 26 = 118.
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