ORIGASMIQUE
Problem setters and judges:
•
•
•
•
•
Christian Colombo
Jean Paul Ebejer
Karl Fenech
Gordon Pace
Chris Porter
Origami: The Art of Paper Folding
Paper Folding Conundrums
• In folding and unfolding a sheet of paper we can mark
lines and intersection points. What points can be
calculated in this manner? Is it possible to trisect an
angle?
• Is it possible to fold a piece of paper is such a manner
that the perimeter of the resulting shape is bigger than
that of the original rectangle?
This Year’s Challenge
• Given:
• A 2x1 sheet of paper; and
• A 2-dimensional shape (the target shape).
This Year’s Challenge
• Given:
• A 2x1 sheet of paper; and
• A 2-dimensional shape (the target shape).
• Expected:
• A sequence of instructions (move/translate the paper, rotate it
around the origin or fold the paper along the x-axis);
• which, if applied to the sheet of paper starting at the origin results in
a shape similar to the target one.
This Year’s Challenge
• Given:
• A 2x1 sheet of paper; and
• A 2-dimensional shape (the target shape).
• Expected:
• A sequence of instructions (move/translate the paper, rotate it
around the origin or fold the paper along the x-axis);
• which, if applied to the sheet of paper starting at the origin results in
a shape similar to the target one.
• More:
• In case of a tie, the solution with the least number of instructions
wins.
• A secondary prize for the fold resulting in the most original shape
will be given.
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
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Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Comparing Solutions
• A grid (per 1/64 units) is overlaid over the target shape
and the given solution.
• Let:
• c be the number of correctly hit points (points both in the solution
and the target shape);
• w be the number of incorrectly hit points (points in the solution but
not in the target shape);
• a be the number of points (the area) of the target shape.
• The score of such a solution is:
Comparing Solutions
• A grid (per 1/64 units) is overlaid over the target shape
and the given solution.
• Let:
• c be the number of correctly hit points (points both in the solution
and the target shape);
• w be the number of incorrectly hit points (points in the solution but
The count of correctly hit
not in the target shape);
points has to be larger than
• a be the number of points (the area)
ofincorrectly
the targethit
shape.
that of
ones
• The score of such a solution is:
Comparing Solutions
• A grid (per 1/64 units) is overlaid over the target shape
and the given solution.
• Let:
• c be the number of correctly hit points (points both in the solution
and the target shape);
• w be the number of incorrectly hit points (points in the solution but
not in the target shape);
The score is never below
zero
• a be the number of points (the area) of the target
shape.
• The score of such a solution is:
Comparing Solutions
• A grid (per 1/64 units) is overlaid over the target shape
and the given solution.
• Let:
• c be the number of correctly hit points (points both in the solution
and the target shape);
• w be the number of incorrectly hit points (points in the solution but
not in the target shape);
• a be the number of points (the area)
of score
the target
shape.
The
is normalised
according to the size of the
• The score of such a solution is:
target shape
Comparing Solutions
• A grid (per 1/64 units) is overlaid over the target shape
and the given solution.
• Let:
• c be the number of correctly hit points (points both in the solution
and the target shape);
• w be the number of incorrectly hit points (points in the solution but
not in the target shape);
Score may range between
0 and 1
• a be the number of points (the area) of the target shape.
• The score of such a solution is:
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
Some Observations
• Most submissions were implementations of either a bounding
box or a convex hull algorithm. There was also a single
machine learning approach.
• Convex hull is intrinsically better than bounding box,
• But still overapproximates shapes that are concave or have holes.
• When the area of these holes exceeds that of the polygons, the
convex hull algorithm drops to zero scores.
The Solutions (the ones who answered)
• 7 Java solutions, 1 C++ solution, 1 C# solution.
• Some teams used a bounding box solution:
• Team #3 (ARIA)
• Team #7 (FusRoHuhs)
• Team #6 (Programoids) – done via a convex hull calculation
• Team #1 (Abstergo) used this for shapes with 6 or more sides
• A common convex-hull implementation was to apply for a
number of times: Take the next edge of the given shape,
translate it to the origin, rotate and fold.
• Team #2 (DCoders) applied this on each edge of the target shape.
• Team #1 (Abstergo) applied this on the convex hull of the target shape,
and performed the step 2*number of sides times.
• Team #8 (Team *) iterated this step until the areas of the target shape
and the generated one matched.
Team #15: Ixaris
• Two algorithms running on multiple threads were
implemented:
1.
2.
Convex hull solution;
A greedy search, adding T;R;F parts to the solution.
Team #16: Crimsonwing
• Three methods were adopted (running on separate threads):
1. Bounding box fitting
2. Genetic algorithm – a sequence of translations and rotations as
the chromosome (no longer than 4*number of edges), with an
implicit fold every three operations.
3. Constraint Programming – on the parameters of the operations.
• After 55 seconds, the best sequence from all the methods is
optimised (removing operations with no parameters, collapses
similar operations, etc.).
• Unfortunately, some bugs in the geometry functions resulted in
inaccurate results.
Team #17: 6PMers
• Our approach was divided into two parts:
• Bounding box calculation
• Find the points intersecting with the bounding box and join them to
find lines corresponding to edges of the original object.
• Rotate and fold accordingly – this was not done due to lack of time.
Instead…
• Re-position/translate it to the required destination without further
operations.
Some Problems and Solutions…
Problem: #7 (The Flattened Cube)
Problem: #7 (The Flattened Cube)
Teams #1, #11, #15
Problem: #7 (The Flattened Cube)
Team #8
Problem: #7 (The Flattened Cube)
Teams #6, #7, #17
Problem: #7 (The Flattened Cube)
Team #2
Problem: #9 (The Learner Plate)
Problem: #9 (The Learner Plate)
Teams #6, #7, #17
Problem: #9 (The Learner Plate)
Team #8
Problem: #9 (The Learner Plate)
Team #11
Problem: #9 (The Learner Plate)
Problem: #16 (The Gents)
Problem: #16 (The Gents)
Team #8, #11, #15
Problem: #16 (The Gents)
Teams #6, #17
Problem: #16 (The Gents)
Problem: #18 (?)
Problem: #18 (?)
Teams #8, #11
Problem: #18 (?)
Teams #6, #17
All the Problems in the Suite
All the Problems in the Suite
All the Problems in the Suite
All the Problems in the Suite
All the Problems in the Suite
Some Unobserved Ideas
• Where the convex hull approach fails due to concavity or holes,
a “single hit” solution, which simply aims to hit a single grid
point would be better. No one implemented this.
• Another possibility for such shapes would have been to identify
a better scoring convex shape than the convex hull. Some
partial solutions to this seem to have been used by some.
• A set of general shape solutions (e.g. V, X, U, O shapes) may
have also been used to generate alternative algorithmic
solutions. No one implemented this.
• No team exploited the tie-breaker:
• If everyone achieves a zero score for a shape, the points would be
awarded to the teams with the least operations.
• … in which case an empty list of instructions is the best solution
The Results: Student Teams
Team
Score
Ops
The Results: Student Teams
Team
Score
Ops
Team #11:
450
15435
Team #7: FusRoHuhs
410
281
Team #2: DCoders
130
1136
The Results: Student Teams
Team
Score
Ops
Team #1: Abstergo
480
1012
Team #11:
450
15435
Team #7: FusRoHuhs
410
281
Team #2: DCoders
130
1136
The Results: Student Teams
Team
Score
Ops
Team #6: Programoids
550
314
Team #1: Abstergo
480
1012
Team #11:
450
15435
Team #7: FusRoHuhs
410
281
Team #2: DCoders
130
1136
The Results: Student Teams
Team
Score
Ops
Team #8: Team *
860
888
Team #6: Programoids
550
314
Team #1: Abstergo
480
1012
Team #11:
450
15435
Team #7: FusRoHuhs
410
281
Team #2: DCoders
130
1136
The Results: Student Teams
Team
Score
Ops
Team #8: Team *
860
888
Team #6: Programoids
550
314
Team #1: Abstergo
480
1012
Team #11:
450
15435
Team #7: FusRoHuhs
410
281
Team #2: DCoders
130
1136
Team #3, #5, #10, #14: No executable or batch file named origasmi.
Team #4, #9, #12, #13: No submission.
The Results: Industry Teams
Team
Score
Ops
The Results: Industry Teams
Team
Score
Ops
Team #15: Ixaris
1030
1689
Team #17: 6PMers
850
247
Team #16: CrimsonWing
All crashed or timed out
Combined Results Winner
Team #8: Team *
The Most Original Fold
The Most Original Fold
Team *’s Ugly Duckling
Are All Solutions Really Possible?
• Until 2002, it was thought that paper could only be folded
a maximum of seven times, due to loss of paper in the
folding process.
• In 2002, a 17 year old high school student Britney
Gallivan proved this belief wrong by successfully folding a
single roll of toilet paper twelve times.
• She used a 1200m long piece of toilet paper.
The Problem of the Problem Problems…
The password to decrypt the file with the judging problems
is (final comma, grammatical, spelling mistakes but not
quotes included):
“Rajt ma rajtx, smajt ma smajtx, qal Duminku fuq l-ghatba ta nanntu,”