AMC 8 · 2020 · #6

Grade 1 logic

Archetype Small-Domain Constrained Search

logical-deductionsystematic-enumeration caseworkif-then-reasoningsystematic-enumeration ↑ Prerequisites: logical-deduction

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Problem

Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?

$\textbf{(A) }\text{Aaron} \qquad \textbf{(B) }\text{Darren} \qquad \textbf{(C) }\text{Karen} \qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}$

Pick an answer.

(A)

$text{Aaron}$

(B)

$text{Darren}$

(C)

$text{Karen}$

(D)

$text{Maren}$

(E)

$text{Sharon}$

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Toolkit + CCSS Solution

Understand

Restated: Five friends — Aaron, Darren, Karen, Maren, Sharon — sit in a five-car train with one seat per car. Cars are numbered $1$ (front) through $5$ (back). Using the clues, figure out who sits in car $3$ (the middle car).

Givens: Maren is in car $5$ (the last car); Aaron sits directly behind Sharon, so wherever Sharon is in car $n$, Aaron is in car $n+1$; Darren is somewhere in front of Aaron, so $\text{car}(D) < \text{car}(A)$; At least one person sits between Karen and Darren, so $|\text{car}(K) - \text{car}(D)| > 1$ (they are not next to each other); Answer choices: (A) Aaron, (B) Darren, (C) Karen, (D) Maren, (E) Sharon

Unknowns: The person who sits in car $3$, the middle car

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #3 Eliminate Possibilities, #1 Draw a Diagram

The most restrictive clue is the $SA$ block (Sharon immediately followed by Aaron), and Maren is already pinned to car $5$. So the only freedom for $S$ and $A$ is which pair of adjacent cars in $1$–$4$ they occupy — that gives exactly three cases. Tool #2 (Systematic List) lays out those three placements in order so none is missed. Tool #3 (Eliminate Possibilities) then tests each case against the Darren and Karen clues and crosses out the ones that break a rule. A small Tool #1 picture (five boxes labeled $1$–$5$) keeps the bookkeeping concrete.

Execute — Answer: A

#1 Draw a Diagram K.G.A.1 Step 1

Draw five empty boxes for cars $1$ through $5$ (front to back) and fill in what is fixed: Maren in car $5$.
Use first initials for everyone else.

$$[\,\_\,,\,\_\,,\,\_\,,\,\_\,, M\,]$$

💡 "In front of" and "behind" are Kindergarten position words — drawing the cars in a row makes those words physical.

#2 Make a Systematic List K.G.A.1 Step 2

Since Aaron sits directly behind Sharon, $S$ and $A$ form an adjacent pair $SA$.
The pair has to fit inside cars $1$–$4$ (car $5$ is Maren).
List every possible starting car for Sharon in order: $1$, $2$, $3$.

Case 1: $[S,A,\_,\_,M]$ \quad Case 2: $[\_,S,A,\_,M]$ \quad Case 3: $[\_,\_,S,A,M]$

💡 Listing the cases in order of Sharon's car number guarantees we don't skip any arrangement.

#3 Eliminate Possibilities 1.NBT.B.3 Step 3

Test Case 1 against the Darren clue.
Aaron sits in car $2$, so Darren needs a car numbered less than $2$ — that is car $1$.
But car $1$ is already Sharon's.
Darren has nowhere to sit, so Case 1 is eliminated.

$\text{car}(D) < 2 \Rightarrow \text{car}(D) = 1$, but car $1 = S$. Contradiction.

💡 Comparing $1$ and $2$ with $<$ is a Grade 1 number-comparison move.

#3 Eliminate Possibilities 1.NBT.B.3 Step 4

Test Case 2.
Aaron is in car $3$, so Darren must be in car $1$ or car $2$ — but car $2$ is Sharon, leaving only car $1$.
Then Karen takes the only remaining seat, car $4$.
Check the spacing clue: there are two people ($S$ in car $2$ and $A$ in car $3$) between Karen and Darren, so $|4-1|=3 > 1$.
Every clue is satisfied.

$$[D, S, A, K, M]\,, \;\; |4-1| = 3 > 1\;\checkmark$$

💡 Checking $|4-1|=3$ and comparing it with $1$ is straightforward Grade 1 subtraction-and-compare.

#3 Eliminate Possibilities 1.NBT.B.3 Step 5

Test Case 3.
Aaron is in car $4$, so Darren can sit in car $1$ or car $2$, and Karen takes the other.
But whichever way you assign them, $K$ and $D$ end up in cars $1$ and $2$, which are adjacent: $|1-2| = 1$, not greater than $1$.
Both sub-cases fail.

$D{=}1, K{=}2: |2-1|=1$ \; ✗ \qquad $D{=}2, K{=}1: |1-2|=1$ \; ✗

💡 If $K$ and $D$ end up next door, the "at least one between" rule breaks — a direct Grade 1 comparison.

#2 Make a Systematic List K.G.A.1 Step 6

Only Case 2 survives, so the unique seating is $[D, S, A, K, M]$.
The middle car (car $3$) holds Aaron.

Middle car $= 3 \Rightarrow$ Aaron $\;\Rightarrow\; \textbf{(A)}$

💡 Reading off the person in the middle position is exactly the Kindergarten "describe positions" idea.

[1] #1 K.G.A.1 Draw five empty boxes for cars $1$ through $5$ (front to back) and fill in what

[2] #2 K.G.A.1 Since Aaron sits directly behind Sharon, $S$ and $A$ form an adjacent pair $SA$.

[3] #3 1.NBT.B.3 Test Case 1 against the Darren clue. Aaron sits in car $2$, so Darren needs a ca

[4] #3 1.NBT.B.3 Test Case 2. Aaron is in car $3$, so Darren must be in car $1$ or car $2$ — but

[5] #3 1.NBT.B.3 Test Case 3. Aaron is in car $4$, so Darren can sit in car $1$ or car $2$, and K

[6] #2 K.G.A.1 Only Case 2 survives, so the unique seating is $[D, S, A, K, M]$. The middle car

Review

Reasonableness: Plug the arrangement $[D, S, A, K, M]$ back into every clue. Maren is in car $5$ ✓. Aaron (car $3$) is directly behind Sharon (car $2$) ✓. Darren (car $1$) is in front of Aaron (car $3$) ✓. Between Karen (car $4$) and Darren (car $1$) sit Sharon and Aaron — two people, more than one ✓. All four clues hold, and Cases $1$ and $3$ were each ruled out by a specific clue, so the answer is unique. Aaron in the middle is consistent.

Alternative: Tool #4 (Matrix Logic) would also work: build a $5 \times 5$ grid of people-by-cars, immediately mark Maren–car$5$ as ✓ (and X out the rest of Maren's row and car $5$'s column), then mark Aaron X in cars $1$ and $5$, Sharon X in cars $4$ and $5$, etc. Each clue erases cells until only one ✓ per row remains. The grid lands on the same arrangement.

CCSS standards used (min grade 1)

K.G.A.1 Describe positions of objects using above, below, beside, in front of (Interpreting the train-car language — "last car", "directly behind", "in front of", "middle car" — as positions in a row of five boxes.)
1.NBT.B.3 Compare two two-digit numbers using symbols (Comparing car numbers ($1$ through $5$) with $<$ and computing $|\text{car}(K) - \text{car}(D)|$ to check the "at least one person between" clue case by case.)

⭐ This AMC 8 problem only needs Grade 1 number comparison ($1 < 2 < 3$) and Kindergarten position words ($\text{in front of}$, $\text{behind}$, $\text{middle}$) — you already know all of these!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 1)

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