AMC 8 · 2023 · #8

Grade 3 arithmetic

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

logical-deductionsystematic-enumerationparity caseworksystematic-enumeration ↑ Prerequisites: logical-deductionmental-arithmetic

📏 Short solution 💡 3 insights

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Problem

Lola, Lolo, Tiya, and Tiyo each play every other player exactly twice, giving six rounds of matches. The table records each player's six results as a string of 1s (win) and 0s (loss). Lola = 111011, Lolo = 101010, Tiya = 010100, and Tiyo is unknown. We need to find Tiyo's six-digit win-loss string from the five answer choices.

Pick an answer.

(A)

$\texttt{000101}$

(B)

$\texttt{001001}$

(C)

$\texttt{010000}$

(D)

$\texttt{010101}$

(E)

$\texttt{011000}$

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Organize Information in More Ways

The table is naturally read row-by-row ("each player's record"), but the trick is to re-read it column-by-column — each column is one round of two simultaneous matches among the four players, so the four entries in any column must contain exactly two 1s (Tool #15). Once we know each column sums to 2, filling in Tiyo's row is just subtraction: column-sum minus the three known entries. We use Tool #3 (Eliminate Possibilities) as a sanity check against the listed choices, and Tool #2 (Make a Systematic List) to walk through the six columns in order without missing any.

1STEP 1

Read the table by columns, not rows: four players make two matches each round, so every column holds exactly two 1s.

column sum = 2 (for every column)

2STEP 2

Go through the columns one by one: Tiyo's entry is two minus the sum of Lola's, Lolo's, and Tiya's entries that round.

Tiyo's column = 2 - (Lola + Lolo + Tiya)

3STEP 3

Applying it to each round in turn gives Tiyo's six entries as 0, 0, 0, 1, 0, 1.

Tiyo = 0, 0, 0, 1, 0, 1

4STEP 4

Reading those digits left to right gives 000101, which is exactly choice (A).

000101 → (A)

Answer

000101

Cross-check by total wins. Across 4 players and 6 rounds with 2 wins per round, the grand total of 1s in the table must be 6 × 2 = 12. Lola has 5, Lolo has 3, Tiya has 2, and our Tiyo = 000101 has 2, summing to 5 + 3 + 2 + 2 = 12. The row total checks, the column constraint holds for every column, and choice (A) is the only one matching all six column equations.

💡Key takeaway

This AMC 8 problem only needs Grade 3 ideas about splitting players into pairs of matches you already know!

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