Problem 18 · 2003 AMC 8
Hard
Logic & Word Problems
careful-counting
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Answer: D — 6 classmates.
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Hint 1 of 2
Translate the words into distance: "friends" are 1 link from Sarah, "friends of friends" are 2 links — so she invites everyone within 2 links.
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Hint 2 of 2
It's faster to count who's NOT invited: the dots that are 3+ links away or not connected to Sarah at all.
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Approach: measure each dot's link-distance from Sarah, then count the far ones
- Rephrase the rule as distance. Sarah's friends are 1 segment away; their friends are 2 segments away. So an invitation reaches everyone at most 2 links from Sarah.
- Sweep outward from Sarah in layers: layer 1 (direct friends), then layer 2 (their friends). Anyone left over — 3+ links away, or in a separate clump with no path to her — is not invited.
- Reading the graph that way: 4 dots sit in disconnected clusters (no path to Sarah), and 2 more are reachable only after 3 links. Those are the uninvited ones.
- 4 + 2 = 6 classmates not invited.
- You'll see this again: "friends within k steps" is a graph-distance question — spreading outward layer by layer (1 link, 2 links, 3 links…) is exactly how you find everything reachable within a fixed number of hops.
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