AMC 8 · 2003 · #18
Grade 4 countingProblem
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: A graph shows $20$ dots, each representing one of Sarah's classmates, with line segments connecting pairs of classmates who are friends. The central oval labeled "Sarah" is connected to her friends. Sarah's guest list consists of (i) every classmate directly connected to her (her friends) and (ii) every classmate connected to one of those friends (friends of friends). How many of the $20$ classmates are not invited?
Givens: $20$ dots total, each representing one classmate; Line segments mark pairs who are friends; Sarah's friends are the dots directly connected to the "Sarah" oval; Friends of friends are dots one more edge away from Sarah; Answer choices: (A) $1$, (B) $4$, (C) $5$, (D) $6$, (E) $7$
Unknowns: The number of classmates who will not be invited to Sarah's party
Understand
Restated: A graph shows $20$ dots, each representing one of Sarah's classmates, with line segments connecting pairs of classmates who are friends. The central oval labeled "Sarah" is connected to her friends. Sarah's guest list consists of (i) every classmate directly connected to her (her friends) and (ii) every classmate connected to one of those friends (friends of friends). How many of the $20$ classmates are not invited?
Givens: $20$ dots total, each representing one classmate; Line segments mark pairs who are friends; Sarah's friends are the dots directly connected to the "Sarah" oval; Friends of friends are dots one more edge away from Sarah; Answer choices: (A) $1$, (B) $4$, (C) $5$, (D) $6$, (E) $7$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems, #16 Change Focus / Count the Complement
The picture is the problem, so Tool #1 (Draw a Diagram) does the heavy lifting: label each dot with its shortest-edge distance from Sarah ($1$, $2$, $3$, or $\infty$ if disconnected). Tool #7 (Identify Subproblems) then splits "not invited" into two clean buckets: dots in a different piece of the graph from Sarah, and dots inside Sarah's piece but too far away (distance $\ge 3$). Tool #16 (Complement) gives a fast cross-check: instead of recounting the not-invited at the end, we can also verify by counting the invited (distance $1$ and $2$) and subtracting from $20$. Both routes must give the same answer.
Execute — Answer: D
4.OA.A.3 Step 1 - Read the edges off the picture and label every dot with its distance from Sarah.
- The drawing connects Sarah directly to $8$ classmates (the dots labeled in the figure $p$, $o$, $d$, $e$, $j$, $k$, $q$, $s$ — Sarah's friends, distance $1$).
- The triangle of $3$ dots in the lower left ($l$-$m$-$n$) is drawn as its own closed loop with no edge to Sarah's part of the graph, and there is one lone dot in the upper-left area ($a$) with no edges at all.
💡 Annotating the given picture with a distance label on each dot turns "who is invited?" into a routine sort.
4.OA.A.3 Step 2 - Find distance-$2$ dots (friends of friends).
- Following each edge from a distance-$1$ dot to a new dot: $d$ touches $b$ and $c$; $e$ touches $f$; $s$ touches $r$ and $t$; $j$ touches $i$.
- That gives the distance-$2$ set $\{b, c, f, r, t, i\}$ with $6$ dots.
- All of these are invited.
💡 Step one edge further from each direct friend; any newly reached dot is a friend-of-a-friend.
3.OA.D.8 Step 3 - Split "not invited" into two subproblems.
- Bucket $A$: dots with no path to Sarah at all.
- Bucket $B$: dots in Sarah's component whose distance from Sarah is $\ge 3$.
💡 Breaking a count into disjoint buckets is the standard "divide and conquer" move.
3.OA.D.8 Step 4 - Count bucket $A$ (disconnected dots).
- The triangle $l$-$m$-$n$ in the lower-left contributes $3$ dots, and the lone dot $a$ in the upper-left contributes $1$.
- Total $A = 4$.
💡 A separate piece of the graph cannot reach Sarah no matter how many edges you walk, so every dot in it is uninvited.
4.OA.A.3 Step 5 - Count bucket $B$ (too far inside Sarah's component).
- The long chain $e$-$f$-$g$-$h$-$i$-$j$ runs through Sarah's network.
- $e$ and $j$ are direct friends (distance $1$); $f$ (next to $e$) and $i$ (next to $j$) are distance $2$.
- The two middle dots $g$ and $h$ are distance $3$ — too far.
- No other dot in Sarah's component is distance $\ge 3$.
- Total $B = 2$.
💡 Walking outward from Sarah, the chain runs out of "close" labels exactly where $g$ and $h$ sit.
3.OA.D.8 Step 6 Add the two buckets to get the answer.
💡 Disjoint buckets add directly — no double-counting to worry about.
4.OA.A.3 Step 7 - Cross-check with the complement.
- Invited dots: distance $1$ has $8$ and distance $2$ has $6$, so $14$ are invited.
- Total dots are $20$, so not invited $= 20 - 14 = 6$.
- Same answer.
💡 Counting invited and subtracting from $20$ is the complement check — it confirms the $6$ found by direct counting.
4.OA.A.3 Read the edges off the picture and label every dot with its distance from Sarah. 4.OA.A.3 Find distance-$2$ dots (friends of friends). Following each edge from a distance 3.OA.D.8 Split "not invited" into two subproblems. Bucket $A$: dots with no path to Sarah 3.OA.D.8 Count bucket $A$ (disconnected dots). The triangle $l$-$m$-$n$ in the lower-left 4.OA.A.3 Count bucket $B$ (too far inside Sarah's component). The long chain $e$-$f$-$g$- 3.OA.D.8 Add the two buckets to get the answer. 4.OA.A.3 Cross-check with the complement. Invited dots: distance $1$ has $8$ and distance Review
Reasonableness: Two independent counts agree on $6$: direct counting of the not-invited ($4$ disconnected $+\; 2$ at distance $3$) and the complement count ($20 - 14$ invited $= 6$). Sanity-check the buckets: $8 + 6 + 4 + 2 = 20$, accounting for every dot exactly once. The answer must be at least $4$ because the disconnected pieces alone contribute $4$, ruling out (A) $1$; it must be less than the total number of dots minus Sarah's $8$ direct friends, ruling out the larger choices unless many friends-of-friends miss as well, which the picture shows is not the case. The middle option $6$ is exactly right, matching choice (D).
Alternative: Tool #16 (Complement) on its own: skip the bucket split and just count the invited. Sarah's $8$ direct friends are visible immediately. For each direct friend, walk one edge to count new dots: $d \to \{b,c\}$, $e \to \{f\}$, $j \to \{i\}$, $s \to \{r,t\}$, and the others ($p, o, k, q$) reach no new dots. That gives $8 + 6 = 14$ invited, so $20 - 14 = 6$ not invited. Same answer (D), reached without ever naming the disconnected pieces.
CCSS standards used (min grade 4)
4.OA.A.3Solve multistep word problems posed with whole numbers using the four operations, and assess the reasonableness of answers (Reading the picture, classifying every dot by distance from Sarah, and combining the bucket counts with addition and subtraction to reach $6$.)3.OA.D.8Solve two-step word problems using the four operations and represent these problems using equations (Splitting "not invited" into the two disjoint buckets $A$ (disconnected) and $B$ (distance $\ge 3$) and adding $|A| + |B| = 4 + 2 = 6$.)
⭐ Label every dot with how many steps it is from Sarah, then sort: distance $1$ and $2$ get invited, everything else does not — this AMC 8 graph puzzle becomes a Grade 4 sort-and-add exercise giving $6$.
⭐ Label every dot with how many steps it is from Sarah, then sort: distance $1$ and $2$ get invited, everything else does not — this AMC 8 graph puzzle becomes a Grade 4 sort-and-add exercise giving $6$.
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