利用不同性質的端點，我們定義了不同的組態，並找出了相應的遞迴關係式。從複雜的遞迴關係式中，我們得到了只與最外端點及結構有關的幾何關係。藉由這些結果，我們得以算出三維漢諾以圖形上所有可能組態的總合I(n) = 4^((4^(n+1)-1)/3)(6(1+1/4^n)+1/2^(n-2))，並且求出該圖形上的冰餘熵S_(HT)(n) = (2/3)ln 2 = 0.462098120373...。

The ice model is a mathematical and physical problem trying to find out a system’s residual entropy related to the system’s special structures, such as fractals with scaling invariance. In this paper, we discuss the ice model on the three-dimensional Hanoi graph and present some relevant exact solutions.
With different kinds of vertices, we can find out different configurations and their corresponding recursive relations, which are associated with the graph’s fractal structures under the ice-rule constrain. By calculating and listing symmetrically these recursive relations, we can obtain profound geometric relations germane to configurations’structures and vertices. The ideal of geometric relations may be an important key to solving other complex ice-model problems. Using these relations, we can present the summation of all possible configurations on the three-dimensional Hanoi graph, I(n) = 4^((4^(n+1)-1)/3) (6(1+1/4^n )+1/2^(n-2)), as well as the exact solution of the residual entropy on the three-dimensional Hanoi graph, S_(HT)(n) = (2/3)ln 2 = 0.462098120373....

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