Triangle Quizes

Toothpick Triangles Math Puzzle

The trick with these puzzles and many more like them is to try to visualize the solution without removing any of the toothpicks. You might think that manipulating the objects themselves would be a good way to go—but surprisingly that’s not always the case. To solve these problems you will need to shift focus from time to time. You might think a puzzle can be solved in a certain way, only to find that you’ll have to take a different, counter-intuitive approach to solve it.

 Puzzle 1Rearrange four toothpicks from the arrangement to make six triangles. | |

The trick here is to visualize each of the four rhombi (plural of rhombus) divided in half. Each rhombus (a quadrilateral with four sides of equal length) divides into two triangles. The arrangement at the left has eight triangles and requires four extra toothpicks. If we remove the green toothpicks, shown in the middle diagram, we get the diagram at the right, with the original number of toothpicks and exactly six triangles. Adding four red toothpicks and removing four green ones is the same as rearranging the four green toothpicks to form six triangles. |

Puzzle 2Remove one toothpick from this arrangement and rearrange the others to form six identical triangles. | |

This arrangement contains 13 toothpicks. When we remove one toothpick, we are left with 12 toothpicks to form six identical triangles. If we look at our solution to problem 1, we see that this indeed uses 12 toothpicks to form six triangles. So we’ve already solved puzzle 2. |

Puzzle 3Arrange the six toothpicks here to form eight equilateral triangles. | |

This requires a shift in our thinking. For the first two puzzles all the triangles had a single toothpick as the side length and all the triangles were the same size. We can easily see from the solution to puzzles 1 and 2 that there is no way we can make eight triangles of this size with only six toothpicks. We’re going to have to overlap the toothpicks...