Solution: 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 1

Rearrange 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 2

Remove 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 3

Arrange 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 in some way. This is one in which trial and error—along with visualization—can help. Let’s start by forming two equilateral triangles.

If we try overlapping the triangles in the same orientation we see that they form shapes other than triangles. But if we turn one of the triangles upside down, we can overlap them to form a six-pointed star with six smaller triangles around the outside. And if we count our two larger triangles, we have a total of eight. (Notice that the problem did not say that the eight triangles have to be the same size.)

Puzzle 4

Rearrange the twelve toothpicks to form seven diamonds.

Having solved puzzle 3, we’ll think about overlapping the shapes right away. We start with three identical diamonds. If we move the two outer diamonds together, overlapping the middle one, we get the shape below. This solves the problem. We have four small diamonds in the middle plus our original three large diamonds, for a total of seven.

Puzzle 5

Using exactly six toothpicks, make four triangles the same size as the ones here.

In a way this is the trickiest of the five puzzles. We have to think even more differently. The four triangles are constructed using nine toothpicks. There does not seem to be any way to make four triangles using only six toothpicks. The trick here is to “think out of the box,”—or more specifically, to think out of the plane. If we imagine using our six toothpicks to make a three-dimensional figure, the problem is immediately solved. The simplest three-dimensional solid is a regular tetrahedron, with six equal edges and four triangular faces.