1988 AJHSME Problems/Problem 16

Problem

Placing no more than one $\text{X}$ in each small square, what is the greatest number of $\text{X}$ 's that can be put on the grid shown without getting three $\text{X}$ 's in a row vertically, horizontally, or diagonally?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

$[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); } [/asy]$

Solution

By the Pigeonhole Principle, if there are at least $7$ $\text{X}$ 's, then there will be some row with $3$ $\text{X}$ 's. We can put in $6$ by leaving out the three boxes in one of the main diagonals.

$\rightarrow \boxed{\text{E}}$

1988 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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1988 AJHSME Problems/Problem 16

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