$1.$ Design a counter with two $$flip flops, and, and one input, $.$If$,$ it counts $1,3,0$ and repeat; if$,$ it counts $1,2,3$ and repeat.

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$8.$a$$ Assume the following two combinations never occur: $(1)$ counter state $2$ and $,$ and $(2)$ counter state $0$ and $($so use don$$t cares $$X$\text{'}$ for these two cases$).$ Give the results for each of the following steps:

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Step $1$: Complete the following state table with state assignment $($AB$=00,$ counting $0$; AB$=01,$ counting $1$; AB$=10,$ counting $2$; AB$=11,$ counting $3)(5$ points$)$

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x

A

B

A$*$

B$*$

$0$

$0$

$0$

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$0$

$0$

$1$

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$0$

$1$

$0$

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$0$

$1$

$1$

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$1$

$0$

$0$

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$1$

$0$

$1$

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$1$

$1$

$0$

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$1$

$1$

$1$

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$$ Step $2$: K$-$map of $$and :$(5$ points$)$

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$$ Step $3$: Flip flop input equations for $($Use the quick method; No circuit needs to be plotted$)$ :$(5$ points$)$

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$8.$b$$ After building the design of part a $($with the two don$$t cares$),$ what happens if somehow is $0$ in state $2$ and what happens if somehow x is $1$ in state $0?$

$(5$ points$)$