Three sticks
Slide to choose three stick lengths, then press Swing the sticks. The longest stick lies flat. The two shorter sticks are hinged to its ends and swing down towards each other. If their tips meet, you have a triangle. If they cannot reach, you see the gap.
What's going on
Lay the longest stick down. The other two have to bridge it, one from each end. Lying flat, end to end, they cover a length equal to their two lengths added together. If that is less than the longest stick, they can never touch, however you swing them.
If the two shorter lengths add up to more than the longest, the tips overlap when flat, so you can lift them until they just meet. That is a triangle. So one test decides every case: the two shorter sticks together must be longer than the longest one.
Try 3, 5 and 8. The tips do touch, but only when both sticks lie flat on the long one. That is a straight line, not a triangle. Now find three lengths where the triangle is very thin.