Problem 1 · 2026 AMC 8
Easy
Arithmetic & Operations
groupingarithmetic-series
What is the value of the following expression?
1 + 2 − 3 + 4 + 5 − 6 + 7 + 8 − 9 + 10 + 11 − 12
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Answer: A — The answer is 18.
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Hint 1 of 2
Don't read it as one long line — notice the signs themselves repeat in a short cycle. What does that cycle tell you about how to chunk the terms?
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Hint 2 of 2
The signs run +, +, − over and over. That's the group-by-the-pattern trick: chop into groups of three and add the group totals instead of the 12 separate terms.
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Approach: let the sign pattern choose the group size
- The signs march +, +, − over and over — that's a clue to chop the 12 terms into groups of three, not to slog left to right.
- Each group is (small + next − bigger). The first few: 1+2−3 = 0, 4+5−6 = 3, 7+8−9 = 6, 10+11−12 = 9. The group totals just climb by 3.
- Add the four group totals: 0 + 3 + 6 + 9 = 18.
- Why this transfers: when a long expression has a repeating sign or operation pattern, group by the length of that pattern — the messy string usually collapses into a short, regular list you can add in your head.
Another way — split into a clean +-sum and a -sum:
- Add every term as if all were positive: 1+2+…+12 = 12×132 = 78.
- The subtracted terms are 3, 6, 9, 12; we had counted them as +, so we must remove them twice: 2 × (3+6+9+12) = 2 × 30 = 60.
- 78 − 60 = 18.
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