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)...
 

What is the Tangram
   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...
  The 7 polygons or 'tans' (see image further below) that form the Tangram are:
      • 5 right triangles: 2 small (hypotenuse of n/2 and sides of n/2√2); 1 medium (hypotenuse of n/√2 and sides of n/2); 2 large (hypotenuse of n and sides of n/√2). The large triangle is 4 times the size of the small triangle, but curiously its perimeter is only 2 times as big!
      • 1 square (side of n/2√2).
      • 1 parallelogram/rhomboid (sides of n/2 and n/2√2).
  Of these 7 pieces, the parallelogram (or rhomboid) is the only piece that may need to be flipped when forming certain shapes; in fact, it has no reflection symmetry but only rotational symmetry, and so its mirror image can only be obtained by flipping it over.

Aim of the puzzle
   The object of the Tangram game is to put the seven geometric shapes together so as to form a given outline/silhouette (rules of the game). Sometimes there is more than one solution. Alternative solutions are accepted as long as they have EXACTLY the same outline of the matching figure. For instance, the two pink figures opposite seem identical but are in fact NOT the same.

First challenge
  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, this one is a classic).

The 7 'Tans' of the Tangram

©1996-2007, Sarcone & Waeber, Genoa
 

  Once you have tried that, see if you can form a larger square using the above 'tans' and adding the extra shape 1) below... Can you do the same using shape 2)? or shape 3)?

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 is 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.

Early Chinese and Western Tangram books

The illustration opposite comes from one of the earliest books of Tangram problems “Nuovo Giuoco Chinese, ossia Raccolta di 364 Figure Geometriche formate con un Quadrato diviso in 7 pezzi” published by Agapito Franzetti (Editor), 1817, Italy. One of the ‘Chinese’ puzzle players is seen cutting his own Tangrams from cardboard, while the two others try to solve a geometric problem.   First Tangram Books printed in Europe and in USA: • “Nuovo dilettevole Giuoco Chinese”, Bertinazzi (Publisher), 1813(?), Italy. • “Nuovo Giuoco Chinese”, Flli. Bettalli (Publisher), 1817, Italy. • “The Fashionable Chinese Puzzle”, John Wallis (Publisher), 1817 USA. • “Enigmes Chinoises”, Grossin (Publisher), 1817, France. • “Metamorfosi del Giuoco detto l'Enimma Chinese”, Landi (Publisher), 1818, Italy.

Two of the first books that talk of the Tangram puzzle published in China around the first decades of the XIXth century.    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?