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?
Triangle has base 4 and height 4
No figure. Cannot copy and paste here.
To determine the solution, let's analyze the figure and triangle pattern:
The figure is a right-angled triangle with a base of 4 toothpicks and height of 4 toothpicks. We can see that there are two smaller triangles placed next to each other, with their bases forming the hypotenuse of the larger triangle.
To find the number of toothpicks required to form the total figure, we can count the number of toothpicks in each triangle and add them together.
For the larger triangle:
- It has a base of 4 toothpicks and height of 4 toothpicks. So, it requires 4 + 4 + 4 = 12 toothpicks.
For the two smaller triangles:
- Each triangle has two equal sides and a hypotenuse formed by the base of the larger triangle.
- A right-angled isosceles triangle with sides of length n can be formed using n + (n-1) + (n-1) toothpicks.
- In this case, each smaller triangle requires 4 + 3 + 3 = 10 toothpicks.
- Since there are two smaller triangles, we need a total of 2 * 10 = 20 toothpicks.
Adding up the toothpicks used in the larger triangle and the two smaller triangles, we have 12 + 20 = 32 toothpicks.
To remove all the triangles from the figure, we need to remove the same number of toothpicks as the total number of toothpicks used in the figure. In this case, that means removing 32 toothpicks. Therefore, the fewest number of toothpicks that could be removed so that no triangles remain is 32.