2022 AMC 8 Problems/Problem 23
Contents
Problem
A 
or 
is placed in each of the nine squares in a 
-by- grid. Shown below is a sample configuration with three s in a line.
![[asy] //diagram by kante314 size(3.3cm); defaultpen(linewidth(1)); real r = 0.37; path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle; draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,1)--(3,1)--(3,2)--(0,2)--cycle); draw((1,0)--(1,3)--(2,3)--(2,0)--cycle); draw(circle((3/2,5/2),1/3)); draw(circle((5/2,1/2),1/3)); draw(circle((3/2,3/2),1/3)); draw(shift(0.5,0.38) * equi); draw(shift(1.5,0.38) * equi); draw(shift(0.5,1.38) * equi); draw(shift(2.5,1.38) * equi); draw(shift(0.5,2.38) * equi); draw(shift(2.5,2.38) * equi); [/asy]](http://latex.artofproblemsolving.com/f/7/9/f79f74a5bed66611fe027a9dd5ab0948b3a033ac.png)
How many configurations will have three s in a line and three s in a line?
Solution 1 (Casework)
Notice that diagonals and a vertical-horizontal pair can never work, so the only possibilities are if all lines are vertical or if all lines are horizontal. These are essentially the same, so we'll count up how many work with all lines of shapes vertical, and then multiply by 2 at the end.
We take casework:
Case 1: 3 lines:
In this case, the lines would need to be 
of one shape and 
of another, so there are 
ways to arrange the lines and ways to pick which shape has only one line. In total, this is 
Case 2: 2 lines:
In this case, the lines would be one line of triangles, one line of circles, and the last one can be anything that includes both shapes. There are 
ways to arrange the lines and 
ways to choose the last line. (We subtract from the last line because one arrangement of the last line is all triangles and the other arrangement of the last line is all circles, which causes Case 2 to overlap with Case 1 and further complicating the solution.) In total, this is 
Finally, we add and multiply: 
.
~wamofan ~further explaining by anonymous
Solution 2
We will only consider cases where the three identical symbols are the same column, but at the end we shall double our answer as the same holds true for rows. There are ways to choose a column with all 's and ways to choose a column with all 's. The third column can be filled in 
ways. Therefore, we have a total of 
cases. However, we overcounted the cases with complete columns of with one symbol and complete column with another symbol. This happens in 
cases. 
. However, we have to remember to double our answer, giving us 
ways to complete the grid.
~MathFun1000
Video Solution
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https://youtu.be/fL7DKXZjmAo?t=239
~ pi_is_3.14
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~Mathematical Dexterity
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~Interstigation
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~David
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~STEMbreezy
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See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 | ||
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