AMC 8 · 2024 · #4

Grade 3 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

perfect-squaresmultiplesmental-arithmetic systematic-enumeration ↑ Prerequisites: multi-digit-arithmeticperfect-squares

📏 Medium solution 💡 3 insights

Problem

Yunji meant to add every integer from 1 to 9, but she accidentally left one of them out. Her resulting (wrong) sum happens to be a perfect square. Among the answer choices (A)~(E), we must identify which integer she left out.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

Everything in this problem lives in a tiny range — there are only 9 possible missing numbers and the wrong sum is squeezed into a short interval. So we just list the perfect squares in order (1, 4, 9, 16, 25, 36, 49, …) until one falls in the allowed range (Tool #2). We can double-check by guessing and checking each answer choice (Tool #6). Once we know the wrong sum is 36, we work backwards from 45 - x = 36 to recover x (Tool #11). No algebraic machinery (Tool #13) is needed for an elementary student.

1STEP 1

Pair from the outside in — 1+9, 2+8, 3+7, 4+6 each make 10, plus the middle 5 — so 1 through 9 sums to 45.

1+2+3+4+5+6+7+8+9 = (1+9)+(2+8)+(3+7)+(4+6)+5 = 10+10+10+10+5 = 45

2STEP 2

With x from 1 to 9, the wrong sum 45 - x lands between 36 and 44 inclusive.

x ∈ {1, 2, …, 9} → 45 - 9 ≤ 45 - x ≤ 45 - 1 → 36 ≤ 45 - x ≤ 44

3STEP 3

Among squares 1, 4, 9, 16, 25, 36, 49, only 36 lands in [36, 44], so the wrong sum is 36.

1, 4, 9, 16, 25, 36_∈ [36,44] ✓, 49, … → wrong sum = 36 = 6²

4STEP 4

Reversing 45 - x = 36 gives x = 45 - 36 = 9 — the number Yunji left out, choice (E).

45 - x = 36 → x = 45 - 36 = 9 → (E)

Answer

Let's double-check directly. If Yunji left out 9, her sum is 1+2+3+4+5+6+7+8 = 36 = 6 × 6 = 6² — indeed a perfect square. The range [36, 44] contains only one perfect square (36), so the answer is unique. We can also rule out the other choices by direct subtraction: 45-5=40, 45-6=39, 45-7=38, 45-8=37 — none of these is a perfect square. So answer (E) 9 is correct.

💡Key takeaway

This AMC 8 problem only needs the Grade 3 times-table fact 6 × 6 = 36 you already know!

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