# 8. Variant 1

Tanya had a set of identical sticks. She formed a large triangle from them, with each side consisting of 11 sticks, and laid out a pattern inside the triangle such that the triangle was divided into smaller triangles with a side of 1 stick (the figure shows an example of such a pattern for a triangle with a side of 3 sticks). How many sticks did Tanya use in total?

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-7.jpg?height=385&width=436&top_left_y=1068&top_left_x=861)
Answer: 198.

Solution (1st method). All the sticks can be divided into those that go horizontally, at an angle to the right, and at an angle to the left:

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-7.jpg?height=212&width=241&top_left_y=1759&top_left_x=926)

In the first such row, there is 1 stick, in the second - 2 sticks, in the third - 3 sticks, and so on. There are 11 rows in total, so there are $3 \cdot(1+2+\ldots+11)=3 \cdot 66=198$ sticks in total.

Solution (2nd method). Notice that all the sticks can be divided into groups of 3, forming a triangle with a side of 1 stick:

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-7.jpg?height=230&width=268&top_left_y=2292&top_left_x=923)

Then, on the first level from the top, there will be one such triangle, on the second level - 2 triangles, and so on. On the bottom level, there will be 11 triangles (because the base of the triangle will consist of 11 sticks). In total, we get: $3 \cdot(1+2+\ldots+11)=3 \cdot 66=198$ sticks.

## Variant 2

Tanya had a set of identical sticks. She formed a large triangle from them, with each side consisting of 17 sticks, and laid out a pattern inside the triangle so that the triangle was divided into small triangles with a side of 1 stick (the figure shows an example of such a pattern for a triangle with a side of 3 sticks). How many sticks did Tanya use in total?

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-8.jpg?height=380&width=439&top_left_y=585&top_left_x=862)

Answer: 459.

## Variant 3

Tanya had a set of identical sticks. She formed a large triangle from them, with each side consisting of 15 sticks, and laid out a pattern inside the triangle so that the triangle was divided into small triangles with a side of 1 stick (the figure shows an example of such a pattern for a triangle with a side of 3 sticks). How many sticks did Tanya use in total?

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-8.jpg?height=379&width=439&top_left_y=1369&top_left_x=862)

Answer: 360.

## Variant 4

Tanya had a set of identical sticks. She formed a large triangle from them, with each side consisting of 19 sticks, and laid out a pattern inside the triangle so that the triangle was divided into small triangles with a side of 1 stick (the figure shows an example of such a pattern for a triangle with a side of 3 sticks). How many sticks did Tanya use in total?

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-8.jpg?height=382&width=439&top_left_y=2156&top_left_x=862)

Answer: 570.

## Variant 5

Tanya had a set of identical sticks. She formed a large triangle from them, with each side consisting of 14 sticks, and laid out a pattern inside the triangle so that the triangle was divided into small triangles with a side of 1 stick (the figure shows an example of such a pattern for a triangle with a side of 3 sticks). How many sticks did Tanya use in total?

![](https://cdn.mathpix.com/cropped/2024_05_06_cd3ddbbeed702155f0e9g-9.jpg?height=377&width=439&top_left_y=522&top_left_x=862)

Answer: 315.