Problem:
A or is placed in each of the nine squares in a grid. Shown below is a sample configuration with three 's in a line.
How many configurations will have three 's in a line and three 's in a line? To determine the number of configurations with three 's in a line and three 's in a line?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
First, there can't be a horizontal line of one shape and a vertical line of another shape. This can't happen as it would require a position to take both shapes. This means we can consider only when the lines are horizontal and only when the lines are vertical. Since rotating the shapes will take the lines from horizontal to vertical, we only need to check how many vertical lines there are as there are equally as many horizontal lines. Now, when finding configurations, we can have separate vertical lines or only vertical lines. We can split this into cases.
Case 1: lines:
There are choices for the first line (which could be all triangles and all circles), choices for the second line, and choices for the third line. This would make choices. However, there are two cases to ignore in which each line is the same. These cases would ensure that only one shape has a line. Therefore, with lines, we have total choices.
Case 2: lines:
Since there are two lines and we need at least one line of each shape, each shape can only have one line. There are ways to choose a spot for the line of triangles and ways to choose a spot for the line of circles. Now, with the remaining spots, we need to ensure that a line isn't created or it would fall into the other case. This would mean we have choices for the remaining line. Totally, we have choices with lines. This means we have configurations with vertical lines. As shown before, we have the same number of horizontal configurations, so we have horizontal configurations. This makes total cases.
Thus, the answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions