AMC 10 · 2023 · #12

Grade 9 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
sum-of-squares-formulasequences-arithmeticpolynomial-factoringexponents pattern-recognitionconvert-to-algebraidentify-subproblems ↑ Prerequisites: exponentssequences-arithmetic
📏 Medium solution 💡 2 insights
Problem
Evaluate a long chain that adds every even cube from two up to eighteen and subtracts every odd cube from one up to seventeen. Report the total.

Pick an answer.

(A)
2023
(B)
2679
(C)
2941
(D)
3159
(E)
3235
How to solve
Strategy Introduce a Variable

Cubing eighteen numbers and alternating the signs works, but it is slow and error-prone. Tool #7 (Identify Subproblems) first cuts the chain into nine couples, each an even cube minus the odd cube one below it. Tool #5 (Look for a Pattern) shows all nine couples share one shape. Then the load-bearing move, Tool #4 (Introduce a Variable): call the couple index k and expand (2k)³ - (2k-1)³ once. The cubic parts cancel and a quadratic survives, so nine cubic differences turn into one quadratic summed nine times. Tool #3 (Eliminate Possibilities) provides an independent divisibility check on the five choices at the end.

1STEP 1

Chop it into pairs

Chop it into nine pairs.

S = (2³ - 1³) + (4³ - 3³) + (6³ - 5³) + … + (18³ - 17³)
2STEP 2

Index the pairs

Give the pairs an index.

S = Σ_k=1⁹ [ (2k)³ - (2k-1)³ ]
3STEP 3

Expand one general pair

Expanding makes the cubes cancel.

(2k-1)³ = 8k³ - 12k² + 6k - 1, so (2k)³ - (2k-1)³ = 8k³ - (8k³ - 12k² + 6k - 1) = 12k² - 6k + 1
4STEP 4

Sum the quadratic

Use the standard sum formulas.

S = 12Σ_k=1⁹ k² - 6Σ_k=1⁹ k + Σ_k=1⁹ 1, where Σ_k=1⁹ k = (9 · 10)/2 = 45, Σ_k=1⁹ k² = (9 · 10 · 19)/6 = 285, and Σ_k=1⁹ 1 = 9
5STEP 5

Finish the arithmetic

The arithmetic gives 3159.

S = 12(285) - 6(45) + 9 = 3420 - 270 + 9 = 3159 → (D) 3159
Answer
3159
Three checks agree. First, the pair formula at both ends: 12k² - 6k + 1 gives 7 at k = 1, and 2³ - 1³ = 8 - 1 = 7; it gives 12(81) - 54 + 1 = 919 at k = 9, and 18³ - 17³ = 5832 - 4913 = 919. The expansion is right where it is easiest to test. Second, divisibility by 9: the three pieces 12 · 285 = 3420, 6 · 45 = 270, and 9 are each multiples of 9, so S must be one too. Digit sums of the choices are 2023 → 7, 2679 → 24, 2941 → 16, 3159 → 18, 3235 → 13, and only 3159 is a multiple of 9 — the check alone pins (D). Third, brute force: the nine pair values are 7, 37, 91, 169, 271, 397, 547, 721, 919, and adding them gives 3159 again.
💡Key takeaway

Two cubes that sit one apart barely differ: (2k)³ - (2k-1)³ collapses from cubic all the way down to 12k² - 6k + 1. Add that over k = 1 to 9 and eighteen cubes shrink to 12(285) - 6(45) + 9 = (D) 3159.

  • Chop the chain into nine pairs
  • Index the pairs by k
  • Expand one general pair
  • Sum the quadratic over nine values
  • Finish the arithmetic

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