Problem 26 · 2012 Math Kangaroo
Hard
Logic & Word Problems
caseworkcareful-counting
David wants to place the twelve numbers from 1 to 12 in a circle so that two adjacent numbers always differ by 2 or 3. Which numbers are therefore adjacent?
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Answer: D — 6 and 8
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Hint 1 of 2
List which numbers each value is allowed to sit next to (differ by 2 or 3).
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Hint 2 of 2
1 and 2 have very few options — those force several neighbours; build out from there.
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Approach: forced neighbours then complete the loop
- Each number's neighbours must differ from it by 2 or 3, and each has exactly two neighbours in the circle.
- 1 can only touch 3 and 4; 2 can only touch 4 and 5; chaining these and continuing yields the loop 6–3–1–4–2–5–7–10–12–9–11–8 back to 6.
- In that loop 8 sits next to 6, so 6 and 8 are adjacent.
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