Tangrams have remained popular for so many years partly because they are so simple and at the same time so complex. In other words, because the individual tans are extremely simple shapes, an almost infinite number of combinations can be derived from them. In fact, there are more than 1 billion possible combinations that can be made with the seven tans [source: Cocchini].
The tans themselves are based on some very basic geometric principles. Each tan can be divided into several component triangles, each one a right isosceles triangle with a hypotenuse equal to √2 units, and two sides that measure 1 unit. (That unit can be inches, centimeters, feet, meters or even a made-up unit, because the shapes are based on proportional, not numerical, measurements).
For example, the small triangles in the set are composed of two base triangles lined up side-by-side. The square is made up of two base triangles joined at the hypotenuse, and so on. To draw a set of tangrams, you can simply draw a square, superimpose a 4x4 grid over it, divide each square into two triangles, and then trace out the shapes along the borders of those triangles so that they match a tangram template. It doesn't matter what units you use to draw the grid, as long as it is perfectly square.
Often, tangram puzzles take on a shape, like a cat, a person or a sailboat. When it comes to these freeform shapes, there are potentially infinite combinations (especially when you factor in nonsense shapes that don't necessarily look like anything). However, there are some mathematical categories of figures that have set rules. These are easier to define and count.
Mathematical figures are those whose base triangles can all be lined up to a square grid. In other words, each shape is aligned so that at least one of its sides is perfectly horizontal or vertical [source: Koller]. With fully matched figures, every tan has at least one of its edges and one of its corners, or vertices, matched to at least one other tan. That is, there aren't any dangling pieces whose outlines can be easily identified. There are also fully aligned figures that can have dangling pieces, but at least one of the edges of any dangling tan has to form a continuous line with the border of the figure [source: Cocchini].
One specific subset of fully matched figures that mathematicians have studied is convex figures. These silhouettes are convex polygons -- shapes with interior angles all less than 180 percent. An easy way to tell if a polygon is convex is to draw a line between any two angles of the shape. If all of those lines either completely fit inside the figure, or perfectly match one of its borders, the shape is convex. Believe it or not, there are only 13 convex polygons that can be made out of the seven tans [source: Wang]. By contrast, the tans can form more than 10 million fully matched shapes [source: Cocchini].