Problem 16 · 2016 Math Kangaroo
Stretch
Logic & Word Problems
casework
Hannes has a game board with 11 spaces (see picture). He places one coin each on eight spaces that lie next to each other. He can choose on which space to place his first coin. No matter where Hannes starts, some spaces will definitely be filled. How many spaces will definitely be filled?
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Answer: D — 5
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Hint 1 of 2
The 8 coins can start at space 1, 2, 3, or 4 (and run forward 8 in a row).
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Hint 2 of 2
Find the spaces that are covered no matter which of those starts is chosen.
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Approach: intersect every possible block of 8 consecutive spaces
- Eight in a row on an 11-space board can begin at space 1, 2, 3, or 4.
- Those blocks are 1–8, 2–9, 3–10, and 4–11; the spaces common to all of them are 4, 5, 6, 7, 8.
- So 5 spaces are always filled.
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