1988 AMC 8 Problem 16
Attempt Problem 16 of the 1988 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1988 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
16.
Placing no more than one X in each small square, what is the greatest number of X’s that can be put on a grid without getting three X’s in a row vertically, horizontally, or diagonally?
Answer: E
Small hint:
Try to leave one square empty in each row and column so no line gets filled
Big hint:
With X’s only two squares are empty, but two empty squares cannot break all three rows, so some row would be full
Solution:
A placement of X’s works: leave the three squares along one main diagonal empty and fill the other six. Then each row and each column is missing one square, the used diagonal has an empty square, and the other diagonal passes through the empty center, so no line of three is complete.
Seven X’s is impossible: only two squares would be empty, and two empty squares can lie in at most two of the three rows, forcing the remaining row to be completely filled with three X’s.
So the greatest number is
Thus, the correct answer is E .
Problem 16 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
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