Problem 21 · 2022 Math Kangaroo
Stretch
Logic & Word Problems
careful-counting
The vertices of a 20-gon are labelled using the numbers 1 to 20 so that adjacent vertices always differ by 1 or 2. The sides of the 20-gon whose vertices are labelled with numbers that only differ by 1 are drawn in red. How many red sides does the 20-gon have?
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Answer: B — 2
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Hint 1 of 2
Going around, the labels almost always jump by 2; jumps of 1 are forced only at the extremes.
Still stuck? Show hint 2 →
Hint 2 of 2
Try odds rising then evens falling and count the steps of size 1.
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Approach: forced arrangement, count step-of-1 edges
- Arranging 1,3,5,...,19,20,18,...,4,2 around the polygon keeps every step at 1 or 2.
- Only the 19–20 edge and the 2–1 edge are steps of exactly 1.
- So there are 2 red sides.
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