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# Magic Triangles, or the Area Paradox

by Gianni A. Sarcone

 "La sapienza è figliola della sperienza. Fuggi i precetti di quelli speculatori, che le loro ragioni non son confermate dalla isperienza" Leonardo da Vinci

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 Cut out a square of 12 units per side into 8 triangular pieces, as shown below (fig.1). We can rearrange these 8 right triangles together to form some square variants. Further below you will find 4 possible configurations. The first configuration is a real square (fig. 2.a); the second one (fig. 2.b), though it seems to be a square is not a square at all! Can you say why this is so? The last examples (fig. 2.c and 2.d) are squares but with extra protruding triangular elements! Compare them to the square of the fig. 2.a). What is wrong?

fig.1
The 8 triangles of the Circea's Puzzle

The 4 possible configurations

(This 8-piece puzzle, called Circea's puzzle and created
by G. Sarcone, is copyrighted: © 1992-2007)

 solution + more exciting vanishing area paradoxes

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