# Tangram: Modern Discoveries and Achievements The main feature of the old classic Tangram is a possibility to create different shapes–people, animals, trees, flowers, buildings, and much more–using just seven pieces. Such shapes are very realistic and easily recognizable objects. Recently, many nice Tangram shapes have been published at Puzzlium. At the same time, Tangram hides many great, pure mathematical tasks in its deep. Say, how many convex shapes can be created of the full set of Tangram pieces. Martin Gardner describes this task solved in 1942 by Fu Traing Wang and Chuan-Chih Hsiung. They proved that Tangram creates exactly 13 different convex shapes. All these shapes are shown in Martin Gardner's book. --- # Tangram: Convex Polygons All the thirteen convex polygons possible to create of Tangram. The next question is about different polyominoes (shapes composed of a set of unit squares) Tangram can produce. It is obvious that only one monomino is possible–it is a basic square formed of the full set of Tangram pieces. There is one domino and only one tetromino (again, it is a square). And there is a big set of octominoes to be done of Tangram out of 369 possible octominoes. Recently, the latter task has been explored and some results have been published by Courtney McFarren. --- # Tangram: Octominoes This and the opposite pages show all 136 octominoes (out of 369 possible octominoes) which can be done of Tangram. That web page states that 134 octominoes out of 369 ones can be made with Tangram, but in fact it shows just 132 different octominoes. Four more octominoes (shown in yellow in the diagram in this page) were discovered by Serhiy Grabarchuk during revisiting this theme. The updated results with all 136 octominoes are presented in this and the opposite pages. It is odd enough that just a handful of octomino shapes were published in different Tangram books and playing sets during the two-century history of Tangram. --- # Tangram: Stars The most recent question about different stars or star-like shapes of Tangram was posed by Serhiy Grabarchuk in 2014. This task as well as the task with polyominoes was performed with help of Puzzlium which proposes a great tool for creating and publishing different shapes of Tangram. A Tangram star is a plain shape whose outline is created of alternately repeating peaks and caves (convex and concave angles). Let’s count as peaks right and acute angles, and call such shapes with N peaks – N-pointed stars. There are stars with 4, 5, 6, 7, 8, 9, and 10 points, which can be created of the full Tangram set. Some of them are symmetric, while most of them are asymmetric, and stars with bigger numbers of points become to look like burrs. Are there stars with more than 10 points to be created of Tangram? Last but not least, there are a few samples of newly created geometric paradoxes with Tangram. In each paradox pair its shapes look almost identical except that one of the shapes has some additional element in it, although for each shape the full set of seven Tangram pieces is used every time. Happy Puzzling! --- **Solutions** All solutions to the above puzzles will be posted at [www.ageofpuzzles.com](http://www.ageofpuzzles.com/).