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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) *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).
- 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, P
^{2}, P^{3}, P^{4}, P^{5}, P^{6}can be called p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, and p_{6}. - We can call p
_{5}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 p
_{6}since it is equivalent to p_{1}, i.e., the end result is the same as performing p_{1}. - The elements {e, p
_{1}, p_{2}, p_{3}, p_{4}} 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
^{-1}P^{-1}Q results in the rotation of tiles 5, 6, and 7. (see Figure 3). This is the first key operation. - Other key operations can be expressed in the same manner.
- For example P
^{-1}QP^{2}Q^{-1}R^{-1}P^{-1}R is the second QP^{-1}Q^{-1}R^{-1}P^{2}RP^{-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 S
_{0}, S_{1}, .. S_{12}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, f
^{2}, g^{2}, fg, gf, gf^{2},f^{2}g, fg^{2}, g^{2}f, and fg^{2}f. - 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.
- 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