Competition · AMC preparation · step 4 of 4
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.
Mark column 4
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 DiagramRule out the 4
Cell (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 PossibilitiesLocate the 2 in the column
Rows 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.
In the fourth column, the digit 2 has only one cell it can go in — the bottom one.
▸ Why?
The three cells above the bottom one in the fourth column are each blocked from being 2, yet the column still has to hold a 2, so the 2 is forced into the bottom cell.
▸ Why?
The fourth column must show each of the digits 1, 2, 3, 4 exactly once, so a 2 has to appear somewhere in it.
▸ Why?
The column's four cells pair up one-to-one with the four digits, so every digit — including 2 — gets its own cell in the column.
▸ Why?
The two empty cells near the top cannot be 2, because each of their rows already shows a 2 and a row may not repeat a digit.
▸ Why?
Each row pairs its four cells one-to-one with the four digits, so a digit already placed in the row cannot take a second cell.
▸ Why?
The remaining upper cell cannot be 2, because it is already filled with a 4 and one cell holds just one digit.
▸ Why?
Each cell is paired with a single digit, so a cell already holding a 4 has no room for a 2.
▸ Why?
A 2 must appear in this column, but three of its four cells now refuse a 2; if the one remaining cell also could not hold the 2, the column would have no room for a digit it is required to contain — which is impossible — so that last cell is forced to take the 2.
Read off the answer
So (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.
- Mark column 4
- Rule out the 4
- Locate the 2 in the column
- Read off the answer
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