## Geometric Puzzle (*)

The following applet allows you to play the "popular" game "15-puzzle". There are 15 square pieces on a square board with 16 small squares, one of these always empty, although in a variable place .

The parts are numbered 1 to 15 and the goal is to sort them. If we assume that in the initial position the empty square is always placed to the right and below ,
there are 15! (= 1.307.674.368.000) options for the starting positions of the 15 pieces. With these configurations, only for a half of them, the puzzle can be solved: these good positions are the ones that correspond to "even permutations" of the pieces numbered by 1, 2, ..., 14, 15. In the puzzle, always appears a good starting position, which allows complete the puzzle; but if you choose an "odd permutations" is led to starting positions in which it is impossible to solve it.

The applet also allows you to play a similar puzzle on surfaces. For example, imagine two vertical sides of a large square glued so as to form a cylinder, which allows to make some movements of pieces, which previously could not have been done
.

Now, several problems arise: is that in some cases (which?) all starting positions
are "good" in the sense that the puzzle can be solved? The answer
to this question and the formulation of more general ones will be added here
(enough) later. However, go trying to solve the puzzle on various surfaces (cylinder,
torus, Möbius strip, Klein bottle, projective plane, cube), starting with
choosing small numbers of rows and columns. If you do not know what is the orientation
of a surface by choosing the Möbius strip and the Klein bottle and some
of the images available, you can start to realize some interesting properties...
You can also choose an image on your computer of your taste and use the applet
to play with it (note that in the right window of the applet, the picture can
look distorted). And if you already have a stereoscopy kit, you can even see
the applet and follow the game in stereoscopic
form (for now, only in Portuguese).

If you have trouble with next applet, click
here.

This applet uses **Javaview**

Translated for Atractor by a

CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

(*) This work was carried out under a grant by FCT - Fundação para a Ciência e a Tecnologia.

Difficulty level: Lower Secondary, Upper Secondary, University