Problem 9 · 2007 AMC 8
Easy
Logic & Word Problems
latin-square-deduction
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square?
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Answer: B — 2.
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Hint 1 of 2
The question asks about one cell, so don't solve the whole puzzle — only follow the row and column that pass through that corner.
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Hint 2 of 2
Sudoku logic: attack the most-filled line first. A cell forbids every digit already in its row or column, so the tightest spots get forced one at a time, and each forced cell tightens the next.
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Approach: force the target cell using only its own column
- Aim at column 4, the column through the corner we want. It already holds a 4 (row 3), so no other cell in it can be 4 — that's the lever.
- Row 1's two blanks are 3 and 4, but column 4 has banned 4, so row 1 col 4 = 3. Same move on row 2 (blanks 1 and 4): col 4 = 1.
- Column 4 now shows 3, 1, 4, leaving only one digit for the bottom-right square: 2.
- Transfers everywhere: in any Latin-square / Sudoku cell, when three of four values are accounted for in a line, the fourth is forced — chase eliminations, not the whole grid.
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