Thirty identical toothpicks were used to create the figure below. There are over 25 triangles in the figure. What is the fewest number of toothpicks that could be removed so that no triangles remain?
The image is an equilateral triangle with base four.
To determine the fewest number of toothpicks that could be removed so that no triangles remain, let's analyze the figure.
The equilateral triangle with base four consists of three triangles. Each side of the equilateral triangle is made up of two toothpicks. Therefore, the total number of toothpicks in the figure is 3 * 2 = 6.
In order to remove all triangles, we need to remove every toothpick that forms a triangle. In the figure, each side of the equilateral triangle has three toothpicks forming a triangle. Hence, the number of toothpicks forming a triangle is 3 * 3 = 9.
Therefore, the fewest number of toothpicks that could be removed so that no triangles remain is 9.
To find the fewest number of toothpicks that could be removed so that no triangles remain in the figure, we first need to count the number of toothpicks in the original figure.
The image shows an equilateral triangle with a base of four. To count the toothpicks, we need to start from the base and count both the horizontal and diagonal toothpicks.
Starting from the bottom, let's count the toothpicks in each row:
- The bottom row has four toothpicks (the base of the equilateral triangle).
- The second row has three diagonal toothpicks.
- The third row has two diagonal toothpicks.
- The top row has one diagonal toothpick in the middle.
Now, let's add up the number of toothpicks in each row: 4 + 3 + 2 + 1 = 10 toothpicks.
So, the original figure has 10 toothpicks.
We're given that there are over 25 triangles in the figure. In an equilateral triangle, each side forms one triangle when combined with the other two sides. So, the number of triangles in the figure can be determined by the number of sides (toothpicks) in the original figure.
Since the equilateral triangle has three sides, the number of triangles in the figure will be a multiple of three. Therefore, we need to find the smallest multiple of three that is greater than 25.
The smallest multiple of three greater than 25 is 27 (3 * 9). This means that if we remove 27 toothpicks, there will be no triangles remaining.
However, the question asks for the fewest number of toothpicks that could be removed. Since 27 is the smallest multiple of three greater than 25, we cannot remove fewer than 27 toothpicks to ensure no triangles remain.
Therefore, the fewest number of toothpicks that could be removed so that no triangles remain is 27.
sorry - still no figure
but google can provide similar problems.
Call a triangle with side length 1 toothpick a 1-triangle. The figure contains 10 upward-pointing 1-triangles and 6 downward-pointing 1-triangles. Removing a toothpick destroys at most one upward-pointing 1-triangle, so we must remove at least 10 toothpicks. Any triangle must have at least one horizontal toothpick, so if we remove 10(answer)
horizontal toothpicks, no triangles remain. Since we have to remove at least 10, this is the minimum.