Consider the unsupervised data of Figure 10.5 (page 477).

(a) How many different stable assignments of examples to classes does the kmeans algorithm find when k = 2? [Hint: Try running the algorithm on the data with a number of different starting points, but also think about what assignments of examples to classes are stable.] Do not count permutations of the labels as different assignments.

(b) Estimate how many different stable assignments there are when k = 3.

(c) Estimate many different stable assignments there are when k = 4.

(d) Why might someone suggest that three is the natural number of classes in this example? Give a definition for “natural” number of classes, and use this data to justify the definition.