Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #4
Grade 3 geometry-2dPick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Add 1 through 9 by pairing
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.
Adding nine single-digit numbers using a pairing trick is Grade 2 addition fluency within 100.
2.NBT.B.5Make A Systematic ListBound the wrong sum
With x from 1 to 9, the wrong sum 45 - x lands between 36 and 44 inclusive.
Using a one-step subtraction within 100 to bound the possible values is a Grade 2 word-problem skill.
2.OA.A.1Guess And CheckFind the square in that range
Among squares 1, 4, 9, 16, 25, 36, 49, only 36 lands in [36, 44], so the wrong sum is 36.
Recognizing 6 × 6 = 36 as a perfect square uses Grade 3 single-digit multiplication fluency within 100.
The incorrect sum can only be 36.
▸ Why?
The incorrect sum is a square number, and out of every value it could possibly take, only 36 is a square number.
▸ Why?
The incorrect sum can only fall between 36 and 44, because it is the full total 45 with one of the numbers 1 through 9 removed.
▸ Why?
The numbers 1 through 9 add up to 45, since pairing 1+9, 2+8, 3+7, 4+6 makes four tens plus a leftover 5, and reordering and regrouping the addends never changes the total.
▸ Why?
The addends may be taken in any order.
▸ Why?
The addends may be grouped in any way before adding.
▸ Why?
The leftover total is the whole 45 split into the removed number and the rest, so removing the smallest number 1 leaves 44 and removing the largest number 9 leaves 36.
▸ Why?
Across the whole stretch from 36 to 44 the only square number is 36, because 6 groups of 6 already reach 36 and the next square, 7 groups of 7, jumps straight to 49.
▸ Why?
A square number is a whole number laid out as that many equal groups of itself, so 6 groups of 6 give 36 and 7 groups of 7 give 49 with nothing in between.
Work backwards to the missing number
Reversing 45 - x = 36 gives x = 45 - 36 = 9 — the number Yunji left out, choice (E).
Solving "what do I subtract from 45 to get 36?" is a Grade 2 unknown-addend / subtraction-within-100 task.
2.OA.A.1Work BackwardsThis AMC 8 problem only needs the Grade 3 times-table fact 6 × 6 = 36 you already know!
- Add 1 through 9 by pairing
- Bound the wrong sum
- Find the square in that range
- Work backwards to the missing number
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