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.