Tangram, the incredible timeless 'Chinese' puzzle

Tangram puzzle, is also called: chinese puzzle - gioco cinese, or rompicapo cinese (it) - casse-tête chinois (fr) - chinesisches Rätselspiel, or japanesisches Legespiel (ger) - chineesch raadsel (du) - 七巧板 qi-qiao ban or 七巧圖 qi-qiao tu (chin) - タングラム or chie-no-ita (jap)...
 

  The Tangram is nowadays the most popular dissection puzzle formed from 7 polygons. The aim of the puzzle is to seamlessly arrange all the geometric pieces to form problem figures (rules of the game). More than 100 years ago, this game was as famous as the Rubik cube and has been played passionately by many as entertainment, educational or mathematical tool, because it boosts shape recognition, problem solving, and pattern design skills. It is said that the Pythagorean theorem was discovered in the Orient with help of Tangram pieces...
  Print this page and cut out the 7 shapes below. Then, try with all 7 pieces to compose a perfect square (there are several solutions).

The 7 Tans of the Tangram

©1996-2007, Sarcone & Waeber, Genoa
 

  Once you have done this, try to fit alternately the shapes 1), 2) and 3) below into your square.

3 additional pieces

(Click on the shapes above to see solutions)

  Little is known for certain about the origin of the Tangram. Even the origin of the name is obscure! The earliest known book was published in 1813 in China, but the publication date is not reliable. Nevertheless, one Tangram-like puzzle first appeared in a book published in Japan in 1742.
  Scholars assume that Tangram began in the Orient before the 18th century and then spread westward. Frankly, in my humble opinion, a lot of 'oriental' games were first created in Europe and then readapted in Asia, like the "Chinese checker", called tiao-qi in China (the "Chinese checker" was actually invented in Germany in 1892 and it's a descendant from the game Halma)... In the past, the adjective 'Chinese' was commonly used to denote any odd, complicate or contrived thing and not the origin! However, by 1817, Tangram publications had appeared in the United States and in Europe. Whatever date the Tangram was invented, you have to know that rearrangement puzzle roots can be traced back to the 3rd century BC! Back in those days, Archimedes, a Greek mathematican, designed a Tangram-like puzzle called Loculus Archimedis or Ostomachion.
  Toward the end of 19th century, Friedrich Ad. Richter, a German industrialist, began to manufacture stone versions of Tangram along with other dissection puzzles under the name of 'The Anchor Puzzle' (Anker). The Anchor puzzles were so successful that over 30 new designs of puzzle sets followed.

     The Tangram Legend.

  Both of the 'squares' below are made from the same 7 tangram pieces. Why are 2 small triangles missing in the second one?