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Magic Triangles, or the Area Paradox (solutions 2)

by Gianni A. Sarcone
Pages: | 1 || 2 || 3 |

Reassembling triangular puzzle pieces, induces always to paradoxical conclusions. The squares of the fig. 2.c) and 2.d) have extra triangular elements. Is then their area larger than the one in the fig. 2.a)? As before, we have to consider the angles of each right triangle which form these squares. By doing that, we will easily notice that the hypotenuse slopes of the small and of the large right triangles are slightly different (a difference of approx. 0.8 degrees, visually unnoticeable). So, the 8 right triangles do not form exactly a square and the sum of all these tiny fitting errors (grey zones in the fig. 5) is equal to the area of the protruding triangular elements. In short, space apparition is only illusion!
example 5
Paradoxical missing square puzzle
(called 'Fehlendes-Quadrat-Puzzle', in German; and 'wigparadox', in Dutch)
We can even enhance the 'vanishing area paradox' effect by adding 4 squares of 6 units per side to the Circea's puzzle. When you rearrange the puzzle pieces a square space appears or disappears (see examples below).

example 6
Both squares are formed with exactly the same 12 pieces.
The second one, however, needs an extra piece!

For classrooms and amateur magicians
Our puzzles Quadrix and Geometrex are based on the above vanishing area principle: when several pieces of the puzzles are permuted, a small square hole magically appears...
You can purchase these hands-on games, useful for classroom demonstrations or for magic close-up shows, online.
A lot of well known magic tricks, such as the Winston Freer Tile Puzzle or the Paul Curry's Geometrical Paradox, are inspired by these basic vanishing principles.

For manufacturers or publishers
We are presently actively looking for manufacturers or publishers interested to produce and market the vanish Quadrix magic vanish puzzle under licence for specific countries on exclusive or non-exclusive basis. Please contact us if interested for details.

Back to page 2
More vanish puzzles here


© 1992-2007 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.
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