# 3 Educational Value

CopyCat is an exciting educational tool. It challenges the mind to understand complex geometrical structures. There are some tools that can aid in understanding as well.

At the lowest level, the player must first break down the picture into its component parts, in order to then recognize those same parts as the faces of the object. Once a required part of the picture is on the object's top face, pattern recognition must be used again to identify the face's proper position in the copy.

One of the first discoveries that can be made is that the law of commutativity is, in general, broken. In other words, one rotation and then a second rotation will present a particular top face; however, starting from the same position, the second rotation followed by the first one may present an entirely different top face.

CopyCat is filled with symmetry. For instance, the cube has a solid white face. This face can be rotated by 90 degrees, 180 degrees, and 270 degrees--and it will still look the same: call this 4-way symmetry. This means that there are four different orientations of the cube that have the white face on top. As well, there are faces on the cube that are not the same as themselves rotated by 90 degrees, but are the same if rotated by 180 degrees: call this 2-way symmetry. Not only do the faces have symmetries, but so do the solid objects. So just playing this game will expose the player to numerous symmetries.

Another interesting point is that the octahedron alternates between the top face as an upward-pointing triangle, and a downward-pointing triangle. So, it would appear that a particular face could have 6 different orientations. As it turns out, this is not the case; each face can have only 3 different orientations from being rolled around. This phenomenon is related to the number of faces that meet at a vertex on the object and the number of faces that meet in the plane.

There is a further complication in level two: the top face will be imprinted on the copy as a mirror-image of itself. This adds the concept of reflection to the game.

With the basics out of the way, how does a player play well? Fortunately, there is direct feed back for the player: the minimum number of rotations needed to copy the picture accompanies the number of rotations the player used. In the first level, the player must find the shortest way (in rotations) to all the faces on the object. At first, the player may rely on the rotating copy of the object, and then later try to use that knowledge on the active-play object. After playing several times, the player will soon become familiar with some of the paths between the faces, as the same problem to be solved is presented numerous times for each puzzle. Remembering the paths to each face may be tougher for the more complicated pictures, so to do well, players may need to develop their own creative strategies to solve the puzzles.

The second level does not display the minimum number of rotations required to solve the puzzle, so players will not know how well they have done. Players may therefore ask themselves, "Is this the best I can do?". My hope is that players will begin to discuss with one another what their scores were, and how they were achieved. In doing so, players would need to develop a way of talking about these geometrical problem. This would further promote the search for some underlying structures.

## Some Subtleties uncovered

(for the really keen)

For a science student in university, the game becomes a matter of writing down the tree (a graph with no loops) using the directions of rolling as the directed edges. The tree can then be used as a map to navigate from the starting position to each of the patterns required. This tree is created by an exhaustive approach. It is not unique. Here's the cube's tree and the octahedron's tree to look at.

In the second level of the program, the object does not start off in the same position for each play. Instead, the object remains in the position of the last move made. This changes the game radically. Now optimally solving the game requires the whole graph of the rollings between positions (vertices). In short, the game becomes "What is the shortest way to travel so that all of the desired vertices are hit?" This would be the same as "the traveling salesman problem," except that the set of vertices that must be visited is not well defined. This problem arises from the equivalencies of certain vertices. For instance, if the solid white face is needed, then there are multiple choices for which white face to use. To say the least, the problem is very tricky--even with the graph with postisions and rolling connections right in front of you. Here's the cube graph and the octahedron graph to look at. Also the newly added truncated octahedron has the same structure as the octahedron (since I do not allow the player to roll on to a square face). But It has nicer tiling than that of the octahedron so it is a nice addition. Here's the truncated octahedron tree and the truncated octahedron graph to look at.

## This cat has claws

The important element of this game is that these difficult problems are carefully hidden in a seemingly elementary environment. I believe that this is what makes the game rich. It is approachable and fun, but it also has content.

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