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Mathematical analysis of Bear Resemblance game

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This puzzle can be solved with a set of key maneuvers
A key maneuver is a sequence of tile movements that result in the exchange of a small number of tiles.
A maneuver, in the general sense, implies transforming any formation into any other formation.
A more restrictive maneuver, which we will call an operation transforms a canonical formation into another canonical formation.
A canonical formation is a formation of the Bear Resemblance game in which the center slot is empty. (see Figure 1)

Figure 1

Elementary operations P, Q, and R, are operations consisting of clockwise rotation of 5 tiles (see Figure 1).
These are short-hand notations for a set of smaller maneuvers. For example, operation P is executed in 6 steps (see Figure 2).

Figure 2

An inverse operation is an operation that cancels out the original operation.
An inverse operation is denoted by the power of negative 1 (e.g. inverse of P is P^-1). (See Figure 2)
An inverse operation can be performed by (but not limited to) executing a maneuver backwards.
A composite operation is two or more operations performed in sequence.
A composite operation is denoted by juxtaposition of operations, and is evaluated from left to right. For example, AB denotes first performing A, then B.
Identical operations performed in sequence n times is denoted by the n-th power. For example, A^2 is the same as AA.
We can call a composite operation an element and give it a name.
For example P, P², P³, P⁴, P⁵, P⁶ can be called p₁, p₂, p₃, p₄, p₅, and p₆.
We can call p₅ an identity element and denote it by the symbol e because it brings the tiles back into the same position, resulting in no exchange of tiles.
We can eliminate the symbol p₆ since it is equivalent to p₁, i.e., the end result is the same as performing p₁.
The elements {e, p₁, p₂, p₃, p₄} form a group of order 5 under the operation of composition.
This group is isomorphic to Z5, which is the group formed by integer addition under modulo 5 arithmetic.
P, Q, R, and all other composite operations thereof constitute an exchange group of a larger order.
The composite operation PQ^-1P^-1Q results in the rotation of tiles 5, 6, and 7. (see Figure 3). This is the first key operation.

Figure 3

Other key operations can be expressed in the same manner.
For example P^-1QP²Q^-1R^-1P^-1R is the second QP^-1Q^-1R^-1P²RP^-1 is the third key operation.
We can combine key operations to form other key operations, which will come in handy when we solve this puzzle.
We will construct a group using the second and third key operation and study its properties.
We can think of Bear Resemblance game as a plastic container that has 13 slots containing 12 tiles.
We name the slots S₀, S₁, .. S₁₂ from left to right, from top to bottom.
Let S_m ® S_n denote the act of moving the content of slot m into slot n.
Let s denote a permutation, i.e., movements of a set of tiles among a closed group of slots.
For example, s : S0 ® S0, S2 ® S6, S4 ® S2, S6 ® S4 is a permutation of 3 tiles within 4 slots.
We use the symbols f and g for the second and third key operations.
We can generate more elements by multiplying f and g arbitrary number of times.
For example, if we first perform f and then perform g, this is f times g, or fg, and it is a distinct element because looking at the permutation, we can tell that it is not equal to either f or g.
There are 12 total unique elements that can be generated this way: e, f, g, f², g², fg, gf, gf²,f²g, fg², g²f, and fg²f.

Figure 4

Figure 4 is a multiplication table.
Figure 5 is a directed graph showing how a new element is generated by multiplication with f or g.

Figure 5

Since the graph is closed, it shows all elements that can be possibly generated.
Figure 6 is a 3-dimentional representaion, emphasizing the symmetrical nature of this graph.

Figure 6

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Last Edit 08/02/2011