Tangrams are a puzzle that originated in China. The earliest known reference to tangrams is from a Chinese book, "The Collected Volume of Patterns of the Seven-Piece Puzzle," which was published in 1803. The puzzle consists of seven geometric pieces.
The pieces are constructed so that the 23 different sides have only four different lengths, two of which are double the length of the other two. In addition, of the 23 corners, there are only two different angles. This leads to hundreds of different puzzles. For example, how many of these tangrams can you construct from the seven pieces?
Actually, one of these tangrams is impossible (I'm not saying which). Can you come up with a convincing argument for why it can never be made with the seven puzzle pieces?
Paradox and Convex Tangrams
Another interesting set of tangrams is the following apparent "paradoxes," each of which can be constructed using all seven puzzle pieces.
Another type of tangram puzzle consists of trying to find all the tangrams with a certain property, such as being convex. A figure is convex if any segment that begins and ends in the figure is contained entirely in the figure. For example, of the six tangrams above, only the triangle and the parallelogram are convex. Can you find all 13 convex tangrams?
The single best source for information on tangrams is "Time Travel and Other Mathematical Bewilderments" by Martin Gardner, W.H. Freeman and Company, 1988. Even if you are not interested in tangrams, anything by Martin Gardner is worth reading.