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Poincare Puzzle
Below is a program that tries to help you understand the Poincare Disk. When you hit "Start", a "curved Triangle" will be selected. Your goal is to pick "generator curve" (red, green, or blue) so that a sequence of reflections around the given generator will land inside the selected one. The reflections start with the main "curved triangle" in the middle. After that, the new triangle will be reflected around one of the colored generators. Try it a couple of times, you'll quickly get the hang of it.
When you click on "Hint", all "curved triangles" will be colored according to a special scheme. The puzzle is to always find the right path of reflections leading to whichever triangle the computer shows you. If you understand the way the reflections work, you will always find the right path immediately, at least when all curvers are colored.