AMC 10 · 2021 · #17

Grade 2 arithmetic

Archetype Small-Domain Constrained Search

logical-deductionset-partitionsystematic-enumerationif-then-reasoning caseworksystematic-enumeration ↑ Prerequisites: logical-deduction

📏 Medium solution 💡 3 insights

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Problem

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- $11,$ Oscar-- $4,$ Aditi-- $7,$ Tyrone-- $16,$ Kim-- $17.$ Which of the following statements is true?

$\textbf{(A) }\text{Ravon was given card 3.}$

$\textbf{(B) }\text{Aditi was given card 3.}$

$\textbf{(C) }\text{Ravon was given card 4.}$

$\textbf{(D) }\text{Aditi was given card 4.}$

$\textbf{(E) }\text{Tyrone was given card 7.}$

Pick an answer.

(A)

Ravon was given card 3.

(B)

Aditi was given card 3.

(C)

Ravon was given card 4.

(D)

Aditi was given card 4.

(E)

Tyrone was given card 7.

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Toolkit + CCSS Solution

Understand

Restated: Five players (Ravon, Oscar, Aditi, Tyrone, Kim) are each dealt $2$ distinct cards from a deck numbered $1$ through $10$, using every card exactly once. Each player's score is the sum of their two cards. Given the scores $11, 4, 7, 16, 17$, figure out which one of the five listed statements about who got which card is true.

Givens: $10$ cards numbered $1$ through $10$, each dealt to exactly one player; Each player gets exactly $2$ cards; Scores: Ravon $= 11$, Oscar $= 4$, Aditi $= 7$, Tyrone $= 16$, Kim $= 17$; Choices: (A) Ravon got $3$, (B) Aditi got $3$, (C) Ravon got $4$, (D) Aditi got $4$, (E) Tyrone got $7$

Unknowns: Which of the five statements is the true one

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #2 Make a Systematic List

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.

Execute — Answer: C

#3 Eliminate Possibilities 1.OA.A.1 Step 1

Oscar has the lowest score ($4$).
Pairs of distinct positive integers summing to $4$: only $\{1, 3\}$ (since $\{2, 2\}$ would reuse a card).
So Oscar holds $1$ and $3$.
Remove them from the pool.

$\text{Oscar} = \{1, 3\}$; pool $= \{2, 4, 5, 6, 7, 8, 9, 10\}$

💡 The smallest score forces the only addition fact: $1 + 3 = 4$.

#2 Make a Systematic List 2.OA.B.2 Step 2

Kim has the highest score ($17$).
Distinct pairs from $\{1, \ldots, 10\}$ summing to $17$: $\{7, 10\}$ and $\{8, 9\}$.
Both are still in the pool, so Kim is one of those two — wait.
Move to Tyrone ($16$) instead: pairs summing to $16$ are $\{6, 10\}, \{7, 9\}$.
Both still available.
Hold off and check Aditi.

Kim: $\{7,10\}$ or $\{8,9\}$; Tyrone: $\{6,10\}$ or $\{7,9\}$

💡 When two scores have ties, jump to the next most-restricted score.

#3 Eliminate Possibilities 1.OA.A.1 Step 3

Aditi has score $7$.
Distinct pairs from $\{1, \ldots, 10\}$ summing to $7$: $\{1, 6\}, \{2, 5\}, \{3, 4\}$.
But $1$ and $3$ are taken by Oscar, so $\{1,6\}$ and $\{3, 4\}$ are out.
Only $\{2, 5\}$ remains.
Aditi holds $2$ and $5$.

$\text{Aditi} = \{2, 5\}$; pool $= \{4, 6, 7, 8, 9, 10\}$

💡 Once Oscar's $1, 3$ are gone, Aditi's only pair adding to $7$ is $2 + 5$.

#3 Eliminate Possibilities 1.OA.A.1 Step 4

Ravon has score $11$.
Distinct pairs summing to $11$: $\{1,10\}, \{2,9\}, \{3,8\}, \{4,7\}, \{5,6\}$.
Cards $1, 2, 3, 5$ are taken, killing $\{1,10\}, \{2,9\}, \{3,8\}, \{5,6\}$.
Only $\{4, 7\}$ is left.
Ravon holds $4$ and $7$.

$\text{Ravon} = \{4, 7\}$; pool $= \{6, 8, 9, 10\}$

💡 Elimination kills four of the five sum-$11$ pairs; only $4 + 7$ survives.

#3 Eliminate Possibilities 1.OA.A.1 Step 5

Now Tyrone ($16$) and Kim ($17$) split $\{6, 8, 9, 10\}$.
Pairs summing to $16$ from this set: $\{6, 10\}$ only ($7 + 9$ needs $7$, taken).
Pairs summing to $17$: $\{8, 9\}$ only ($7 + 10$ needs $7$).
So Tyrone $= \{6, 10\}$ and Kim $= \{8, 9\}$.
Everyone is assigned.

Tyrone $=\{6,10\}$, Kim $=\{8,9\}$

💡 Two players, two pairs, one assignment each — forced.

#3 Eliminate Possibilities 1.OA.A.1 Step 6

Final assignment: Oscar $\{1, 3\}$, Aditi $\{2, 5\}$, Ravon $\{4, 7\}$, Tyrone $\{6, 10\}$, Kim $\{8, 9\}$.
Check each statement: (A) Ravon got $3$ — false (Oscar got $3$).
(B) Aditi got $3$ — false.
(C) Ravon got $4$ — TRUE.
(D) Aditi got $4$ — false.
(E) Tyrone got $7$ — false (Ravon got $7$).
Answer: $(C)$.

$$\textbf{(C) Ravon got card 4}$$

💡 Read off the only statement that matches the unique assignment.

[1] #3 1.OA.A.1 Oscar has the lowest score ($4$). Pairs of distinct positive integers summing to

[2] #2 2.OA.B.2 Kim has the highest score ($17$). Distinct pairs from $\{1, \ldots, 10\}$ summin

[3] #3 1.OA.A.1 Aditi has score $7$. Distinct pairs from $\{1, \ldots, 10\}$ summing to $7$: ${

[4] #3 1.OA.A.1 Ravon has score $11$. Distinct pairs summing to $11$: ${1,10}, {2,9}, {3,8\

[5] #3 1.OA.A.1 Now Tyrone ($16$) and Kim ($17$) split $\{6, 8, 9, 10\}$. Pairs summing to $16$

[6] #3 1.OA.A.1 Final assignment: Oscar $\{1, 3\}$, Aditi $\{2, 5\}$, Ravon $\{4, 7\}$, Tyrone $

Review

Reasonableness: Verify each player's pair sums to the right score: $1+3=4\checkmark, 2+5=7\checkmark, 4+7=11\checkmark, 6+10=16\checkmark, 8+9=17\checkmark$. Every card from $1$ to $10$ appears exactly once. Total $4+7+11+16+17 = 55 = 1+2+\cdots+10$. Everything balances, and statement (C) is the unique truth.

Alternative: Tool #4 (Matrix Logic) — build a $5 \times 10$ grid (players $\times$ cards). Mark Oscar's row: only $1, 3$ can stay because that's the only pair summing to $4$. Cross those cards out of every other player's row. Repeat: Aditi's row only has $2, 5$ left for sum $7$; cross out elsewhere. Repeat for Ravon. The grid yields the same forced assignment without needing to think about it as elimination chains.

CCSS standards used (min grade 2)

1.OA.A.1 Solve addition and subtraction word problems within 20 (Finding pairs of distinct positive integers that sum to a target score ($4, 7, 11, 16, 17$) — all sums within $20$.)
2.OA.B.2 Fluently add and subtract within 20 using mental strategies (Quickly enumerating two-number addition facts ($\{6,10\}$ vs $\{7,9\}$ for $16$, etc.) and cross-checking against the remaining card pool.)

⭐ This 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 2)

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