How to Play the Sierpinski Triangle Game

The Sierpinski Triangle Game is based on The Chaos Game, a java applet created by Johanna Voolich and Robert Devaney at http://math.bu.edu/DYSYS/applets/chaos-game.html.

Upon choosing the Sierpinski Triangle Game from the File/New menu (F9), you will be presented with the second iteration of the Sierpinski Triangle. This consists of nine subtriangles, one of which is colored red. This is the target triangle. You will also see a blue dot at one of the three corners of the large triangle. The goal of the game is to move the blue dot into the interior of the red target triangle (so not just on the boundary, but actually inside the triangle.)

You move the blue dot by choosing one of the three corners (labeled 1, 2, 3). Each time you click on one of the corner buttons (or press the corresponding number key on the keyboard) you will move the blue dot from its current location half the distance towards that particular corner (each corner is a fixed point for one of the functions). The "Goal" label shows you the minimum number of moves that are needed to get the blue dot from its original position into the interior of the red triangle. You win the game if you can determine an appropriate choice of moves to reach the target using the minimum number required. Your sequence of moves is displayed in the Results frame and a yellow dot is left behind when the blue dot moves to its new location. If the dot is hard to see, you can increase the dot size using the + (plus) key, or decrease it using the – (minus) key.

If you change your mind you can undo your previous move(s). To start over completely without changing the target triangle or the starting location of the blue dot, the Try Again (ctrl-A) option from the Draw menu. To try a different target and/or initial location, choose the New Game (ctrl-N) option from the draw menu.

Choosing a different level will change the number of subtriangles obtained in the iterations that define the Sierpinski triangle. For example, level 5 uses 35 = 243 triangles. The goal for the minimum number of moves needed to get the blue dot from its starting corner into the interior of the target triangle will change for each level.

The default game uses the basic Sierpinski iterated function system that consists of three functions, each scaling by 1/2. The game can be modified by choosing to add either one or three rotations from the Rotations menu. If you choose one rotation, you will see a green square representing the fixed point for that function. The way the blue dot moves when corners 1 or 2 are chosen remains the same. However, if you pick corner 3, the blue dot first moves half the distance towards the green square and then rotates about the green square by 180 degrees to the exact opposite side. The initial shape in this game is a four-sided parallelogram and the blue dot can start at any one of the four vertices.

The three rotation game has three green squares (each representing a fixed point). In this game, every move consists of moving the blue dot from its present location half the distance towards the corresponding green square followed by a rotation of 180 degrees around the green square to the opposite side. The initial shape in this game is a six-sided polygon and the blue dot can start at any of the six vertices or at the midpoint of any of the six sides.

One rotation game

Three rotation game