= Tangram =

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== Problem Description ==

Tangram is an ancient Chinese game of Shapes. The goal is to form a particular shape from a set of seven pieces consisting of 5 triangles (of three different sizes), a square, and a parallelogram. These seven pieces are defined below, where the 1-st argument is the block number (an integer identifier),
the 2-nd argument is the number {{{k}}} of sides of the block, 
and the last {{{2*k}}} arguments represents the vertices of the block.
{{{
size_block(1,3, 0,0, 4,0, 2,2). %%% Large triangle 
size_block(2,3, 0,0, 4,0, 2,2). %%% Large triangle 
size_block(3,3, 0,0, 2,0, 0,2). %%% Medium triangle
size_block(4,3, 0,0, 2,0, 1,1). %%% Small triangle 
size_block(5,3, 0,0, 2,0, 1,1). %%% Small triangle 
size_block(6,4, 0,0, 1,-1, 2,0, 1,1). %%% Square 
size_block(7,4, 0,0, 1,1,  1,3, 0,2). %%% Parallelogram 
}}}
Note that these facts are not part of the input.

Allowed angles are 0 (0), 45 (1), 90 (2), 135 (3), 180 (4), 225 (5), 270 (6), and 315 (7) degrees,
represented by means of the numbers 0..7 in parenthesis.

== Predicates ==

 * '''Input''': {{{sides/1, coord/3}}}

 * '''Output''': {{{put/4}}}

== Input format ==

A fact of the form {{{sides(n)}}}, fixing the number of sides of the figure.

The target shape is specified by means of {{{n}}} facts of the form
{{{
coord(1,X1,Y1). 
coord(2,X2,Y2). 
...             
coord(n,Xn,Yn). 
}}}
In particular, a fact of the form {{{coord(i,Xi,Yi)}}} specifies that
the i-th vertex of the target shape is {{{(Xi,Yi)}}}.
Thus, the perimeter of the figure is the set of segments {{{ { (X1,Y1)-(X2,Y2), (X2,Y2)-(X3,Y3), ..., (Xn-1,Yn-1)-(Xn,Yn), (Xn-Yn)-(X1,Y1) } }}}.

In our setting, each coordinate X is an integer in {0,...,8},
and each coordinate Y is an integer in {0,...,6}.

== Output format ==

Your program needs to compute a predicate put/4 that associates to each block i in {1,...,7} 
one pair (X,Y) and one angle A. With that association the entire surface of the instance figure
must be covered (and without overlapping, but this is a consequence for the chosen figures).
More specifically, an atom of the form {{{put(i,x,y,r)}}} in the output specifies that
the block with ID {{{i}}} is positioned in {{{(x,y)}}} and rotated according to {{{r}}}.

== Example(s) ==

Input: {{{sides(3). coord(1,0,0). coord(2,8,0). coord(3,4,4).}}}

Output: {{{put(1,4,0,0). put(2,4,4,6). put(3,4,0,2). put(4,0,0,0). put(5,3,1,2). put(6,1,1,0). put(7,3,1,0).}}}

== Author(s) ==
 * Author:  Agostino Dovier 	
   * Affiliation: University of Udine, Italy
 * Author: Andrea Formisano 	
   * Affiliation: University of Perugia, Italy
 * Author: Enrico Pontelli	
   * Affiliation: New Mexico State University, USA