Competition · AMC preparation · step 4 of 4
AMC 10 · 2021B · #17
Grade 2 arithmeticPick an answer.
AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #3 (Eliminate) is the natural fit: each player's score has only a small number of two-card decompositions, and the most restrictive scores (the lowest, 4, and the highest, 17) have only one decomposition each. Start with the unique forced cards, eliminate those from the pool, then watch the next player's options collapse. Tool #2 (Systematic List) supports each step by enumerating all {a, b} with a + b = score from the remaining pool. Once every player's pair is determined, just read off which of the five statements is true.
Pin down Oscar's cards
Oscar's 4 is the lowest score and splits only as 1 + 3, so Oscar holds cards 1 and 3; drop them from the pool.
The smallest score forces the only addition fact: 1 + 3 = 4.
1.OA.A.1Eliminate PossibilitiesNarrow Kim's options
Kim's 17 and Tyrone's 16 each split two ways still, so defer them and attack a more forced score first.
When two scores have ties, jump to the next most-restricted score.
2.OA.B.2Make A Systematic ListPin down Aditi's cards
Aditi's 7 could be 1+6, 2+5, or 3+4, but 1 and 3 are gone, leaving only 2 and 5 for Aditi.
Once Oscar's 1, 3 are gone, Aditi's only pair adding to 7 is 2 + 5.
1.OA.A.1Eliminate PossibilitiesPin down Ravon's cards
Ravon's 11 has five pairs, but 1, 2, 3, 5 are taken, so only 4 and 7 survive for Ravon.
Elimination kills four of the five sum-11 pairs; only 4 + 7 survives.
Once the earlier cards are taken, elimination kills all but one of the remaining pairs.
▸ Why?
Each card belongs to exactly one player, so a used card is gone from every other pair.
▸ Why?
With every other candidate ruled out, the single survivor is forced.
Split the last four cards
From leftover 6, 8, 9, 10: Tyrone's 16 is 6 and 10, Kim's 17 is 8 and 9 — everyone is assigned.
Two players, two pairs, one assignment each — forced.
1.OA.A.1Eliminate PossibilitiesRead off the answer
Final deal — Oscar 1,3; Aditi 2,5; Ravon 4,7; Tyrone 6,10; Kim 8,9 — so the one true statement is (C).
Read off the only statement that matches the unique assignment.
1.OA.A.1Eliminate PossibilitiesThis AMC 10 problem only needs Grade 2 addition facts you already know! Start with the easiest score — Oscar's 4 only comes from 1+3. Then knock out those cards from everyone else's options; the puzzle collapses one player at a time. Ravon ends up with 4 and 7, so statement (C) is the true one.
- Pin down Oscar's cards
- Narrow Kim's options
- Pin down Aditi's cards
- Pin down Ravon's cards
- Split the last four cards
- Read off the answer
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