Competition · AMC preparation · step 4 of 4
AMC 8 · 2023 · #8
Grade 3 arithmeticPick an answer.
AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Read the table by columns
Read the table by columns, not rows: four players make two matches each round, so every column holds exactly two 1s.
4 players grouped into pairs of opponents gives 2 matches per round — basic Grade 3 multiplication/division reasoning.
In every round of the tournament exactly two of the four players win, so each column of the table adds up to 2.
▸ Why?
In one round all four players are busy at once, and splitting four players two-to-a-match makes exactly two matches happening side by side.
▸ Why?
Two matches with two players each use up all four players, so four players share out into exactly two matches of two.
▸ Why?
Each of those two matches ends with exactly one winner, so the round hands out exactly two wins in all.
▸ Why?
Each match is tied to its own single winner, and two matches paired one-for-one with their winners means two winners.
▸ Why?
A column holds the four players' results for that round as 1s and 0s, and adding the column just counts the winners, which is 2.
▸ Why?
Every loser writes a 0, and adding a 0 changes nothing, so the column total is only the count of the players who wrote a 1.
Set up the column rule
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.
Subtracting a known sum from 2 to find the missing addend is a Grade 1 addition/subtraction-within-20 idea.
1.OA.A.1Make A Systematic ListFill in each column
Applying it to each round in turn gives Tiyo's six entries as 0, 0, 0, 1, 0, 1.
Each column needs the missing addend to make the sum 2 — Grade 1 unknown-addend thinking, repeated six times.
1.OA.A.1Make A Systematic ListAssemble Tiyo's row
Reading those digits left to right gives 000101, which is exactly choice (A).
Matching a built-up answer string against the five choices is straightforward Grade 1 comparison.
1.OA.A.1Eliminate PossibilitiesThis AMC 8 problem only needs Grade 3 ideas about splitting players into pairs of matches you already know!
- Read the table by columns
- Set up the column rule
- Fill in each column
- Assemble Tiyo's row
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