Suppose we have five sticks, each one a different integer length from 1 cm to 5 cm, and we want to make a triangle. There are lots of ways to do it, right?

Let's start by seeing how many triangles we can make with the 1-cm stick included.

After trying a few possible combinations, we might start to be more doubtful about the number of triangles we can possibly make. None of the following combinations seem to work:

The lengths shown make constructing a triangle impossible because one side is longer than (or equal to) the other two put together. In fact, there are *zero ways* to make triangles with these sticks that include the 1-cm stick! Let's look at various combinations with the 3-cm stick to see all the different ways this can fail:

The generalized form of this observation about side lengths in triangles is called the **triangle inequality theorem**:

The sum of the lengths of any two sides of a triangle must be larger than the length of the third side.

With this idea in hand, we can quickly describe why the triangles with the 1-cm stick were never going to work: none of the second-longest sticks can be made longer than the longest by adding 1 cm to their length.

In fact, if we work through the lengths above systematically, we find—somewhat surprisingly—that the only triangles possible with the five sticks given are 2-3-4, 2-4-5, and 3-4-5:

The triangle inequality is one of those statements in mathematics that unlocks a new kind of understanding—once you've grasped the idea behind it, you can use it to solve more complex problems.