How to properly solve this Hidden Markov Model problem?
The Viterbi algorithm solves the problem. It calculates the most probable state sequence for an HMM given an observation sequence. In your example, ring sizes are observations and temperatures are hidden states. I suggest to refer to the Wikipedia article about the Viterbi algorithm (http://en.wikipedia.org/wiki/Viterbi_algorithm). It contains an example of an HMM with 2 hidden states and observations with 3 discrete probabilities.