AMC 10 · 2021 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

order-of-operationsexponentsmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: order-of-operations

📏 Short solution 💡 1 insight

Problem

Evaluate the arithmetic expression (2² - 2) - (3² - 3) + (4² - 4) and match it to one of five choices.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is three small pieces of the same shape n² - n glued together with +, -, +. Tool #7 (Subproblems) says: compute each piece on its own, then combine. Tool #5 (Pattern) is a sanity helper — noticing n² - n = n(n-1) gives the products 2 · 1, 3 · 2, 4 · 3, which makes each subproblem a one-step multiplication.

1STEP 1

Exponent first: 2² = 4, so the first piece is 4 - 2 = 2.

2² - 2 = 4 - 2 = 2

2STEP 2

Same recipe: 3² = 9, so the middle piece is 9 - 3 = 6.

3² - 3 = 9 - 3 = 6

3STEP 3

Again: 4² = 16, so the third piece is 16 - 4 = 12.

4² - 4 = 16 - 4 = 12

4STEP 4

Combine with +, -, +: 2 - 6 + 12 = -4 + 12 = 8, choice (D).

2 - 6 + 12 = -4 + 12 = 8 → (D)

Answer

Each piece is tiny (2, 6, 12), and only the middle one is subtracted. Roughly: 12 + 2 = 14 minus the 6 in the middle leaves 8. That matches choice (D), and it falls between the smallest and largest choices, which is reasonable.

💡Key takeaway

This AMC 10 problem only needs Grade 6 "square the small number, subtract, then add them up" — three quick pieces (2, 6, 12) glued with +, -, + give 8.

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