### Wuxing and Compatibility Coefficient: Part III

If the *xing*were to be arranged in a clockwise manner according to Dong's numerical order, we have the following pentagon of

Interestingly, pentagon is the only polygon for which number of constructive relationship (number of sides) equals the number of destructive relationship (number of diagonals). This is easily proven since the only solution to the following equation:

is*xing*without having to refer to Dong's pentagon.

The three numerical values of compatibility coefficient

For example, in the case of wood*xing*and fire*xing*, we have*xing*and water*xing*, we have

*wuxing*.Interestingly, pentagon is the only polygon for which number of constructive relationship (number of sides) equals the number of destructive relationship (number of diagonals). This is easily proven since the only solution to the following equation:

is

*n*= 5. Now, it is useful for a computer programmer to introduce a coefficient which can be used to compute the compatibility of any two givenThe three numerical values of compatibility coefficient

*c*(namely -1, 0, 1) are to be interpreted as destructive, neutral, and constructive, respectively. A convenient way to compute_{p}*c*is to use the following equation:_{p}For example, in the case of wood

*c*(0,1) = 1, a constructive relationship. In the case of fire_{p}*c*(1,4) = -1, a destructive relationship._{p}
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