AMC 10 · 2009 · #16

Grade 7 arithmetic

Archetype Branching State Enumeration └ Sub-archetype Numeric Operation Tree

absolute-valuesystematic-enumeration caseworksystematic-enumeration ↑ Prerequisites: absolute-value

📏 Medium solution 💡 2 insights

Problem

Four real numbers a, b, c, d sit on a number line. Neighbors are a fixed distance apart: a and b are 2 apart, b and c are 3 apart, c and d are 4 apart. The gap from a to d, measured as |a - d|, can land on several different numbers depending on which way each neighbor jumps. Add up every distinct value that |a - d| can take.

Pick an answer.

(A)

$\ 9$

(B)

$\ 12$

(C)

$\ 15$

(D)

$\ 18$

(E)

$\ 24$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The distance from a to d is built from three separate jumps, and each jump can point either forward or backward. That is a small, finite set of choices: two directions for each of three jumps, so eight cases in total. The clean move is to write a - d as one signed sum of the three jumps, then list all eight sign patterns without missing or repeating any. Once the list is complete, take absolute values, throw out duplicates, and add what remains.

1STEP 1

Chain the gaps together

Rebuild the a-to-d gap from its neighbors: write a - d as (a - b) + (b - c) + (c - d), where b and c cancel in pairs.

a - d = (a - b) + (b - c) + (c - d)

2STEP 2

Turn each absolute value into a sign choice

Absolute value fixes size, not direction, so a - b is ±2, b - c is ±3, c - d is ±4, and the chain gives a - d = ±2 ±3 ±4.

a - d = ± 2 ± 3 ± 4

3STEP 3

List all eight sign patterns

Three independent sign choices make 2 x 2 x 2 = 8 cases, and running them in order gives a - d = 9, 1, 3, -5, 5, -3, -1, -9.

a - d ∈ {9, 1, 3, -5, 5, -3, -1, -9}

4STEP 4

Take absolute values and drop duplicates

Sizes pair up as 9, 1, 3, 5, 5, 3, 1, 9, so the distinct ones are 1, 3, 5, 9 and their sum is 18.

|a - d| ∈ {1, 3, 5, 9}, 1 + 3 + 5 + 9 = 18

Answer

The largest |a - d| happens when all three jumps point the same way: 2 + 3 + 4 = 9, so no value can exceed 9, and the four distinct sizes 1, 3, 5, 9 all sit at or below it. They also all share the parity of 2 + 3 + 4 = 9 (odd), which is why only odd sizes appear and even choices like the answer 12 or 24 never show up as a single |a - d|. Summing the genuine distinct values 1, 3, 5, 9 gives 18, matching (D).

💡Key takeaway

When distances chain together, each hop can go two ways, so list every sign pattern, take absolute values, and add only the different sizes.

Chain the gaps together
Turn each absolute value into a sign choice
List all eight sign patterns
Take absolute values and drop duplicates

Chain the gaps together

Turn each absolute value into a sign choice

List all eight sign patterns

Take absolute values and drop duplicates

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