Different patterns can be created by shading exactly three of the nine small triangles shown, no two of which can share a side.

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-049.jpg?height=235&width=263&top_left_y=1151&top_left_x=888)

Patterns that can be matched by rotations or by reflections are considered the same. For example, the following patterns are considered the same.
![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-049.jpg?height=228&width=746&top_left_y=1530&top_left_x=649)

How many different patterns can be created?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12
Care is needed in systematically counting the different patterns.

Dividing the patterns into groups having some like attribute is one way to help this process.

We will count patterns by grouping them according to the number of "corner" triangles $(3,2,1$, or 0 ) that each has shaded.

## 3 shaded corners

There is only one pattern having all 3 corners shaded, as shown in Figure 1.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=174&width=168&top_left_y=447&top_left_x=1531)

2 shaded corners

To start, we fix the 2 corners that are shaded (the top and bottom right) since rotations will give the other two possibilities that have 2 shaded corners.

For the purpose of identifying the smaller triangles, they have been numbered 1 to 6 as shown in Figure A.

We need only shade one of these 6 numbered triangles to complete a pattern.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=157&width=173&top_left_y=710&top_left_x=1518)

Figure A

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=149&width=155&top_left_y=912&top_left_x=1359)

Figure 2

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=152&width=163&top_left_y=908&top_left_x=1515)

Figure 3

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=163&width=176&top_left_y=905&top_left_x=1685)

Figure 4

Since triangles 2 and 6 each share a side with a shaded corner, they cannot be shaded.

Shading triangle 1 gives our first pattern in this group, as shown in Figure 2.

Triangles 3 and 4 are then shaded to give Figures 3 and 4, respectively.

Note that shading triangle 5 gives a reflection of Figure 3 and so is not a different pattern.

These 3 patterns shown cannot be matched by rotations or reflections.

Thus, there are 3 different patterns having 2 shaded corners.

## 1 shaded corner

Again we start by fixing the corner that is shaded (the top) since rotations will give the other two possibilities that have 1 shaded corner.

We identify the smaller triangles by number in

Figure B.

We need to shade 2 of these 6 numbered triangles to complete a pattern.

Since triangle 6 shares a side with the shaded corner, it cannot be shaded.
![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-235.jpg?height=390&width=696&top_left_y=1574&top_left_x=1256)

Also, no two adjacent triangles can be shaded since they share a side.

Shading triangles 1 and 3 gives our first pattern in this group, as shown in Figure 5.

Triangles 1 and 4 are then shaded to give Figure 6.

Figure 7 has triangles 1 and 5 shaded, while Figure 8 has triangles 2 and 4 shaded.

Shading triangles 2 and 5 gives a reflection of Figure 6 and so is not a different pattern.

Shading triangles 3 and 5 gives a reflection of Figure 5 and so is not a different pattern.

These 4 patterns shown cannot be matched by rotations or reflections.

There are no other combinations of 2 triangles that can be shaded.

Thus, there are 4 different patterns having 1 shaded corner.

No shaded corners

We identify the smaller triangles by number in Figure C.

We need to shade 3 of these 6 numbered triangles to complete a pattern.

Since no two adjacent triangles can be shaded there are only two possible patterns.

Triangles 1, 3 and 5 can be shaded (Figure 9) or triangles 2, 4 and 6 can be shaded (Figure 10).

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-236.jpg?height=165&width=181&top_left_y=194&top_left_x=1514)

Figure C

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-236.jpg?height=201&width=345&top_left_y=382&top_left_x=1432)

These 2 patterns shown cannot be matched by rotations or reflections.

There are no other combinations of 3 triangles that can be shaded.

Thus, there are 2 different patterns having no shaded corners.

Since each of the 4 groups of patterns above has a different number of corner triangles shaded, there is no pattern in any one group that can be matched to a pattern from another group by rotations or reflections.

Therefore, the total number of patterns that can be created is $1+3+4+2=10$.

ANSWER: (C)