AMC 10 · 2016 · #24

Grade 7 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint

sequences-arithmeticdigit-decompositioncasework systematic-enumeration ↑ Prerequisites: digit-decomposition

📏 Long solution 💡 4 insights

Problem

We look at four-digit numbers abcd with a not zero, and read off three two-digit numbers from neighboring digits: ab, bc, and cd. We want these three to be strictly increasing and to form an arithmetic sequence (equal gaps). Count how many four-digit numbers work.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Name the jumps between consecutive digits, turn the equal-gap rule into one short equation, use a multiple-of-9 fact to cut everything down to two cases, then list the members of each case by hand.

1STEP 1

Write the two-digit numbers

By place value each two-digit number is 10·(tens)+(ones); leading digits a, b, c run 1 to 9, only d may be 0.

ab = 10a+b, bc = 10b+c, cd = 10c+d

2STEP 2

Equal gaps equation

Arithmetic means a same gap between terms, so the jump ab→bc must equal bc→cd; that equality is the master condition.

(10b+c)-(10a+b) = (10c+d)-(10b+c)

3STEP 3

Name the digit jumps

Set x=b-a, y=c-b, z=d-c; the two gaps 10x+y and 10y+z are equal, which simplifies to z = 10x - 9y.

x=b-a, y=c-b, z=d-c → 10x+y = 10y+z → z = 10x-9y

4STEP 4

Use a multiple of 9

z-x = 9(x - y) is a small multiple of 9, so x-y is -1, 0, or 1; x-y=1 needs a jump of 9 (impossible), leaving x-y=0 and x-y=-1.

z-x = 9(x-y) → x-y ∈ {-1,0,1}; x-y=1 → z=x+9 ＞ 8 (impossible)

5STEP 5

Case x = y: digits in AP

x=y makes all jumps equal, so a,b,c,d form an AP: step 1 → six, step 2 → three, step ≥3 none, giving 6 + 3 = 9.

x-y=0: a,b,c,d in AP; step 1{:} 6, step 2{:} 3, step ≥ 3{:} 0 → 9

6STEP 6

Case x - y = -1

x-y=-1 gives y=x+1, z=x-9; valid digits allow only x=0,1,2,3, yielding 1,2,3,2 numbers (e.g. 4692) — 1 + 2 + 3 + 2 = 8.

x-y=-1: y=x+1, z=x-9; x=0{:} 1, x=1{:} 2, x=2{:} 3, x=3{:} 2 → 8

7STEP 7

Add the cases

The two cases are disjoint and exhaustive, so add: 9 + 8 = 17 four-digit numbers — choice (D).

9 + 8 = 17

Answer

Every answer choice is small, so a complete count is realistic. The two cases give 9 + 8 = 17, which is choice (D), and the worked example 4692 shows up correctly as one of the x - y = -1 numbers (x = 2).

💡Key takeaway

Turn the words into one equation about the jumps between digits, then a multiple-of-9 trick shrinks endless options into two short lists you can finish by hand.

Write the two-digit numbers
Equal gaps equation
Name the digit jumps
Use a multiple of 9
Case x = y: digits in AP
Case x - y = -1
Add the cases