Difference between revisions of "1988 AJHSME Problems/Problem 16"
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==Problem== | ==Problem== | ||
| − | Placing no more than one <math>\text{X}</math> in each small square, what is the greatest number of <math>\text{X}</math>'s that can be put on the grid shown without getting three <math>\text{X}</math>'s in a row vertically, horizontally, or diagonally? | + | Placing no more than one <math>\text{X}</math> in each small [[square]], what is the greatest number of <math>\text{X}</math>'s that can be put on the grid shown without getting three <math>\text{X}</math>'s in a row vertically, horizontally, or diagonally? |
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math> | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math> | ||
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==See Also== | ==See Also== | ||
| − | + | {{AJHSME box|year=1988|num-b=15|num-a=17}} | |
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 22:22, 3 June 2009
Problem
Placing no more than one 
in each small square, what is the greatest number of 's that can be put on the grid shown without getting three 's in a row vertically, horizontally, or diagonally?
![[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]](http://latex.artofproblemsolving.com/1/1/6/1161c2f219124fd219c171f81bebebb89efeb9a7.png)
Solution
By the Pigeonhole Principle, if there are at least 
's, then there will be some row with 
's. We can put in 
by leaving out the three boxes in one of the main diagonals.
See Also
| 1988 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
