Competition · AMC preparation · step 4 of 4

AMC 8 · 2020 · #6

Grade 1 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

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

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Problem

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).

Pick an answer.

(A)

Aaron

(B)

Darren

(C)

Karen

(D)

Maren

(E)

Sharon

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

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.

1STEP 1

Draw five cars and place Maren

Draw five boxes for cars 1–5 (front to back) and place the one fixed fact: Maren in car 5.

[ _ , _ , _ , _ , M ]

2STEP 2

Pair Sharon with Aaron

Sharon-then-Aaron is an adjacent pair SA that must fit in cars 1–4 (car 5 is Maren); list Sharon's start car in order: 1, 2, 3.

Case 1: [S,A,,,M] Case 2: [,S,A,,M] Case 3: [,,S,A,M]

3STEP 3

Test case 1

Case 1 [S,A,,,M]: Aaron in car 2 needs Darren in front, but the only earlier car is car 1 — Sharon's — so Case 1 is eliminated.

car(D) ＜ 2 → car(D) = 1, but car 1 = S. Contradiction.

4STEP 4

Test case 2

Case 2 [,S,A,,M]: Darren must take car 1, Karen car 4; |4-1|=3 ＞ 1, so [D, S, A, K, M] satisfies every clue.

[D, S, A, K, M] , |4-1| = 3 ＞ 1 ✓

5STEP 5

Test case 3

Case 3 [,,S,A,M]: Karen and Darren must fill cars 1 and 2, which are adjacent (|1-2|=1), so both sub-cases fail.

D=1, K=2: |2-1|=1 ✗ D=2, K=1: |1-2|=1 ✗

6STEP 6

Read the middle car

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

Middle car = 3 → Aaron → (A)

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

All four clues can be satisfied at the same time by only one seating of the five friends, so the person in the middle car is completely determined.

▸ Why?

Because one friend is fixed in the last car and each car seats just one person, Sharon together with the friend seated right behind her — a block filling two neighboring cars — can only begin at the first, second, or third car, leaving just three seatings to check.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

Among those three placements, two are forced to break a clue and one keeps every clue, so exactly one seating survives.

▸ Why?

If the block begins at the first car, the friend behind Sharon takes the second car and the only seat ahead of him, the first, is already Sharon's; since one car cannot hold two people, Darren — who must ride somewhere ahead — has no seat, so this placement fails.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

If the block begins at the third car, the two remaining friends fill the only seats left, the first and second cars, which are neighbors with no car in the gap between them; the whole space between the two friends is empty, so no one can sit between them and this placement fails.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

If the block begins at the second car, the friend who must ride ahead is forced into the one free front car and the last friend into the one remaining car; with their cars far enough apart, every clue holds, so this placement works.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

These three placements together cover every legal seating with none left out, so once the two failing placements are removed, the remaining one must be the true seating.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

In that single surviving seating, four friends already fill four different cars, so the fifth friend and the middle car are left with only one possible choice.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

K.G.A.1Make A Systematic List

Answer

Aaron

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.

💡Key takeaway

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

Draw five cars and place Maren
Pair Sharon with Aaron
Test case 1
Test case 2
Test case 3
Read the middle car

A parent dashboard for the family lives at sensimlab.com.

Draw five cars and place Maren

Pair Sharon with Aaron

Test case 1

Test case 2

Test case 3

Read the middle car

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