AMC 8 · 2007 · #9
Grade 4 logic
Pick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Cell (4,4) lives in column 4. By the rule, column 4 must contain all four digits 1, 2, 3, 4. Three of its cells are either filled or share a row with one of the digits. Tool #3 (Eliminate Possibilities) lets us cross off three candidates for (4,4) until only one digit is left. Tool #1 (Draw a Diagram) keeps a clean picture of the grid so we can see which row and column conflicts apply to each cell. No algebra is needed — only careful row/column bookkeeping.
Draw the grid and focus on column 4; the cell we want is (4,4).
Putting the grid on the page makes the row/column rule visible — each row needs the four digits exactly once, and so does each column. This is the Grade 3 pattern-recognition move.
3.OA.D.9Draw A DiagramCell (3,4) already holds the 4, so (1,4), (2,4) and (4,4) must take the digits 1, 2, 3 in some order.
Removing the digit already placed in the column shrinks the candidate set for every empty cell to {1, 2, 3}.
4.OA.C.5Eliminate PossibilitiesRows 1 and 2 already show a 2, so (1,4) ≠ 2 and (2,4) ≠ 2; with (3,4) filled, the only home for 2 is (4,4) = 2.
Process of elimination on a single column: cross off every cell where the digit cannot go, and the one cell left wins. The digit 2 has only one legal home in column 4.
4.OA.C.5Eliminate PossibilitiesSo (4,4) = 2, which is answer choice (B).
Only one digit survived elimination, so the answer is determined — choice (E) "cannot be determined" is ruled out.
4.OA.C.5Eliminate PossibilitiesFocus on the column that holds the unknown cell. Cross off every digit blocked by its row or by another column entry, and the one digit left is your answer.