Homework 12 Quick Answers

1. For example,pi2 is (a b c d)

2. pi2 and pi3

3. 1/4[2**64 + 2*2**16 + 2**32]

4. n even: 1/2(3*2n-1); n odd 1/2(3*2n-1 + 3*2(n-1)/2))

5. 1/9[3**N + 5*3 + 2*3**3]

6. In the style of burnside’s lemma pick a transformation and look for subsets that are invariant under the transformation. Look at the transformation cycle structure. For a subset to be invariant, every cycle must be uniform with respect to subset: all elements in the subset, or all not in the subset. If there are -say - k cycles in all in the transformation, then there are 2**k ways of building a subset (and its complement) to yield an invariant.

7. a): 1/6[(b+w)6 +(b6 + w6)+2(b3 + w3)**2 + (b2 + w2)**3] ... b) 1/9[(b+w)9 +6(b9 + w9)+2(b3 + w3)**3]

8. for no two colors adjacent, you must have colors in the pattern 2-2-1. 3-1-1, for example, will put two colors next to each other. So generate the pat inventory and select the 2-2-1 patterns, and distinguish between the consecutive ones and the non- consecutive ones. for each pattern there's only one, so: b2w2r + b2wr2 + bw2r2

-- RobbieMoll - 2012-05-07

Topic revision: r3 - 2014-12-04 - RobbieMoll
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