Problem 30 · 2024 Math Kangaroo
Stretch
Logic & Word Problems
work-backwardsum-constraint
A game board is composed of 8 squares on which we want to stack coins. Initially, all squares are empty. On each turn we choose four adjacent squares and place one coin on each of those squares. The numbers show how high the stacks are. Unfortunately, the table wobbled and five of the stacks fell over (shown by ★). How many coins were on the field indicated with a question mark before the stack fell?
| ★ | 30 | 42 | ★ | ★ | 36 | ? | ★ |
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Answer: A — 24
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Hint 1 of 3
Each move adds 1 to a block of four neighbouring squares, so differences between nearby stacks reveal hidden move counts.
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Hint 2 of 3
Let \(m_i\) be how many moves start at square \(i\) (covering squares \(i\) to \(i+3\)); write each known stack as a sum of the \(m_i\).
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Hint 3 of 3
The \(?\) stack at square 7 equals \(m_4+m_5\), which you can pull out of the known stacks 2, 3 and 6.
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Approach: express each stack as a sum of move-counts and subtract
- Let \(m_i\) be how many moves start at square \(i\), so square \(p\)'s height is the sum of all windows covering it.
- Square 2 gives \(m_1+m_2=30\) and square 3 gives \(m_1+m_2+m_3=42\), so \(m_3=12\).
- Square 6 gives \(m_3+m_4+m_5=36\), hence \(m_4+m_5=36-12=24\).
- Square 7's height is exactly \(m_4+m_5\), so the \(?\) stack held 24 coins (answer A).
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