Description
2 Background
The game of tangram consists in creating shapes out of pieces. We assume that each piece has its own
colour, different to the colour of any other piece in the set we are working with. Just for reference, here
is the list the colours that are available to us (you will not make use of this list):
https://www.w3.org/TR/2011/REC-SVG11-20110816/types.html#ColorKeywords
A representation of the pieces will be stored in an .xml file thanks to a simple, fixed syntax.
2.1 Pieces
Here is an example of the contents of the file pieces A.xml, typical of the contents of any file of this
kind (so only the number of pieces, the colour names, and the various coordinates can differ from one
such file to another—we do not bother with allowing for variations, in the use of space in particular).
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 50 50 L 50 90 L 90 90 z " fill =" red "/ >
< path d =" M 160 170 L 160 130 L 120 130 z " fill =" green "/ >
< path d =" M 200 30 L 180 30 L 180 50 L 220 50 z " fill =" blue "/ >
< path d =" M 40 100 L 40 140 L 60 140 L 60 120 z " fill =" yellow "/ >
< path d =" M 210 70 L 230 90 L 270 90 L 270 50 L 230 50 z " fill =" purple "/ >
< path d =" M 180 130 L 180 170 L 220 210 L 240 190 z " fill =" olive "/ >
< path d =" M 100 200 L 120 180 L 80 140 L 80 180 z " fill =" magenta "/ >
Opened in a browser, pieces A.xml displays as follows:
Note that the coordinates are nonnegative integers. This means that the sets of pieces we consider rule
out those of the traditional game of tangram, where √
2 is involved everywhere…
2
We require every piece to be a convex polygon. An .xml file should represent a piece with n sides
(n ≥ 3) by an enumeration of n pairs of coordinates, those of consecutive vertices, the first vertex being
arbitrary, and the enumeration being either clockwise or anticlockwise.
The pieces can have a different orientation and be flipped over. For instance, the file pieces AA.xml
whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 50 50 L 50 90 L 90 90 z " fill =" red "/ >
< path d =" M 160 170 L 160 130 L 120 130 z " fill =" green "/ >
< path d =" M 200 30 L 180 30 L 180 50 L 220 50 z " fill =" blue "/ >
< path d =" M 40 100 L 40 140 L 60 140 L 60 120 z " fill =" yellow "/ >
< path d =" M 210 70 L 230 90 L 270 90 L 270 50 L 230 50 z " fill =" purple "/ >
< path d =" M 180 130 L 180 170 L 220 210 L 240 190 z " fill =" olive "/ >
< path d =" M 100 200 L 120 180 L 80 140 L 80 180 z " fill =" magenta "/ >
and which displays as
represents the same set of pieces (the fact that the latter appear as smaller than the former is just due
to the different scaling of the included pdf’s; the sizes of the pieces are actually the same in terms of the
coordinates of their vertices).
The pieces can overlap, but that does not concern us. In practice, we will just use representations
where the pieces do not overlap as that allows us to visualise the pieces properly when we open the
corresponding.xml file, but it is just for convenience and irrelevant to the tasks we tackle.
3
2.2 Shapes
A representation of a shape is provided thanks to an .xml file with the same structure, storing the
coordinates of the vertices of just one polygon.
The file shape A 1.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 30 20 L 110 20 L 110 120 L 30 120 z " fill =" grey "/ >
and which displays as
is such an example. The file shape A 2.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg ">
< path d =" M 50 10 L 90 10 L 90 50 L 130 50 L 130 90 L 90 90 L 90 130 L 50 130 L 50 90 L 10 90 L 10 50 L 50 50 z" fill =" brown "/ >
and which displays as
is another such example.
Contrary to pieces, shapes are not assumed to be convex polygons. Still they are assumed to be simple
polygons (the boundary of a simple polygon does not cross itself; in particular, it cannot consist of
at least 2 polygons that are connected by letting two of them just “touch” each other at one of their
vertices—e.g., two rectangles such that the upper right corner of one rectangle is the lower left corner of
the other rectangle; that is not allowed).
Whereas you will have to check that the representation of the pieces in an .xml file satisfies our constraints,
you will not have to do so for the representation of a shape; you can assume that any shape we will be
dealing with satisfies our constraints.
4
2.3 Tangrams
The first shape can be built from our set of pieces, in many ways. Here is one, given by the file
tangram A 1 a.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 30 60 L 30 20 L 70 20 z " fill =" green "/ >
< path d =" M 50 120 L 50 80 L 70 60 L 90 80 L 90 120 z " fill =" purple "/ >
< path d =" M 30 100 L 90 40 L 70 20 L 30 60 z " fill =" olive "/ >
< path d =" M 70 60 L 110 100 L 110 60 L 90 40 z " fill =" magenta "/ >
< path d =" M 50 120 L 30 120 L 30 100 L 50 80 z " fill =" blue "/ >
< path d =" M 110 20 L 70 20 L 110 60 z " fill =" red "/ >
< path d =" M 110 100 L 110 120 L 90 120 L 90 80 z " fill =" yellow "/ >
and which displays as follows.
Here is another one, given by the file tangram A 1 b.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 30 20 L 50 20 L 50 60 L 30 40 z " fill =" yellow "/ >
< path d =" M 50 60 L 50 20 L 90 20 L 90 60 L 70 80 z " fill =" purple "/ >
< path d =" M 70 120 L 110 80 L 110 120 z " fill =" green "/ >
< path d =" M 90 20 L 110 20 L 110 40 L 90 60 z " fill =" blue "/ >
< path d =" M 30 120 L 30 80 L 70 120 z " fill =" red "/ >
< path d =" M 70 120 L 30 80 L 30 40 L 90 100 z " fill =" olive "/ >
< path d =" M 70 80 L 110 40 L 110 80 L 90 100 z " fill =" magenta "/ >
and which displays as follows.
5
The second shape can also be built from our set of pieces, in many ways. Here is one, given by the file
tangram A 2 a.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 30 60 L 30 20 L 70 20 z " fill =" green "/ >
< path d =" M 50 120 L 50 80 L 70 60 L 90 80 L 90 120 z " fill =" purple "/ >
< path d =" M 30 100 L 90 40 L 70 20 L 30 60 z " fill =" olive "/ >
< path d =" M 70 60 L 110 100 L 110 60 L 90 40 z " fill =" magenta "/ >
< path d =" M 50 120 L 30 120 L 30 100 L 50 80 z " fill =" blue "/ >
< path d =" M 110 20 L 70 20 L 110 60 z " fill =" red "/ >
< path d =" M 110 100 L 110 120 L 90 120 L 90 80 z " fill =" yellow "/ >
and which displays as follows.
Here is another one, given by the file tangram A 2 b.xml whose contents is
< svg version ="1.1" xmlns =" http :// www . w3 . org /2000/ svg " >
< path d =" M 30 20 L 50 20 L 50 60 L 30 40 z " fill =" yellow "/ >
< path d =" M 50 60 L 50 20 L 90 20 L 90 60 L 70 80 z " fill =" purple "/ >
< path d =" M 70 120 L 110 80 L 110 120 z " fill =" green "/ >
< path d =" M 90 20 L 110 20 L 110 40 L 90 60 z " fill =" blue "/ >
< path d =" M 30 120 L 30 80 L 70 120 z " fill =" red "/ >
< path d =" M 70 120 L 30 80 L 30 40 L 90 100 z " fill =" olive "/ >
< path d =" M 70 80 L 110 40 L 110 80 L 90 100 z " fill =" magenta "/ >
and which displays as follows.
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3 First task (3 marks)
You have to check that the pieces represented in an .xml file satisfy our constraints. So you have to check
that each piece is convex, and if it represents a polygon with n sides (n ≥ 3) then the representation
consists of an enumeration of the n vertices, either clockwise or anticlockwise. Here is the expected
behaviour of your program.
$ python3
Python 3.5.1 ( v3 .5.1:37 a07cee5969 , Dec 5 2015 , 21:12:44)
[ GCC 4.2.1 ( Apple Inc . build 5666) ( dot 3)] on darwin
Type ” help ” , ” copyright ” , ” credits ” or ” license ” for more information .
>>> from tangram import *
>>> file = open ( ’ pieces_A . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
True
>>> file = open ( ’ pieces_AA . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
True
>>> file = open ( ’ incorrect_pieces_1 . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
False
>>> file = open ( ’ incorrect_pieces_2 . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
False
>>> file = open ( ’ incorrect_pieces_3 . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
False
>>> file = open ( ’ incorrect_pieces_4 . xml ’)
>>> coloured_pieces = available_coloured_pieces ( file )
>>> are_valid ( coloured_pieces )
False
Note that the function are valid() does not print out True or False, but returns True or False.
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4 Second task (3 marks)
You have to check whether the sets of pieces represented in two .xml files are identical, allowing for
pieces to be flipped over and allowing for different orientations. Here is the expected behaviour of your
program.
$ python3
Python 3.5.1 ( v3 .5.1:37 a07cee5969 , Dec 5 2015 , 21:12:44)
[ GCC 4.2.1 ( Apple Inc . build 5666) ( dot 3)] on darwin
Type ” help ” , ” copyright ” , ” credits ” or ” license ” for more information .
>>> from tangram import *
>>> file = open ( ’ pieces_A . xml ’)
>>> coloured_pieces_1 = available_coloured_pieces ( file )
>>> file = open ( ’ pieces_AA . xml ’)
>>> coloured_pieces_2 = available_coloured_pieces ( file )
>>> are_identical_sets_of_coloured_pieces ( coloured_pieces_1 ,
… coloured_pieces_2 )
True
>>> file = open ( ’ shape_A_1 . xml ’)
>>> coloured_pieces_2 = available_coloured_pieces ( file )
>>> are_identical_sets_of_coloured_pieces ( coloured_pieces_1 ,
… coloured_pieces_2 )
False
Note that the function identical sets of coloured pieces() does not print out True or False, but
returns True or False.
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5 Third task (3 marks)
You have to check whether the pieces represented in an .xml file are a solution to a tangram puzzle
represented in another .xml file. Here is the expected behaviour of your program.
$ python
Python 3.5.1 ( v3 .5.1:37 a07cee5969 , Dec 5 2015 , 21:12:44)
[ GCC 4.2.1 ( Apple Inc . build 5666) ( dot 3)] on darwin
Type ” help ” , ” copyright ” , ” credits ” or ” license ” for more information .
>>> from tangram import *
>>> file = open ( ’ shape_A_1 . xml ’)
>>> shape = available_coloured_pieces ( file )
>>> file = open ( ’ tangram_A_1_a . xml ’)
>>> tangram = available_coloured_pieces ( file )
>>> is_solution ( tangram , shape )
True
>>> file = open ( ’ tangram_A_2_a . xml ’)
>>> tangram = available_coloured_pieces ( file )
>>> is_solution ( tangram , shape )
False
Note that the function is solution() does not print out True or False, but returns True or False.
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