AMC 8 · 2020 · #6
Easy mode Grade 1Problem
Picture a small train with five cars in a row. Each car holds exactly one person. Call them car 1 (the front), car 2, car 3 (the middle), car 4, and car 5 (the last).
Five people ride this train: Aaron, Darren, Karen, Maren, and Sharon. Here is what we know about where they sit:
- Maren sits in the last car.
- Aaron sits in the car right behind Sharon.
- Darren sits somewhere in front of Aaron.
- Karen and Darren are not in cars next to each other. At least one person sits between them.
Who sits in the middle car?
Pick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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
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
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.
💡 "In front of" and "behind" are Kindergarten position words — drawing the cars in a row makes those words physical.
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: $1$, $2$, $3$.
💡 Listing the cases in order of Sharon's car number guarantees we don't skip any arrangement.
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.
💡 Comparing $1$ and $2$ with $<$ is a Grade 1 number-comparison move.
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.
💡 Checking $|4-1|=3$ and comparing it with $1$ is straightforward Grade 1 subtraction-and-compare.
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.
💡 If $K$ and $D$ end up next door, the "at least one between" rule breaks — a direct Grade 1 comparison.
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.
💡 Reading off the person in the middle position is exactly the Kindergarten "describe positions" idea.
K.G.A.1 Draw five empty boxes for cars $1$ through $5$ (front to back) and fill in what K.G.A.1 Since Aaron sits directly behind Sharon, $S$ and $A$ form an adjacent pair $SA$. 1.NBT.B.3 Test Case 1 against the Darren clue. Aaron sits in car $2$, so Darren needs a ca 1.NBT.B.3 Test Case 2. Aaron is in car $3$, so Darren must be in car $1$ or car $2$ — but 1.NBT.B.3 Test Case 3. Aaron is in car $4$, so Darren can sit in car $1$ or car $2$, and K 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.1Describe 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.3Compare 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!
⭐ 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!
More like this
Same archetype — closest grade level first.