Problem $8.(30$ points$)$ Three children $$ Anna, Bj$$orn$,$ and Cathy $$ are playing in the mud.

They are honest, logical$,$ can see each other$$s faces, but cannot see their own. Let A$,$ B$,$ and C be

propositional variables that represent whether the face of Anna, Bj$$orn$,$ and Cathy, respectively, is

muddy.

The children are told, by their parents, that they must come home if their faces are muddy.

They are quite obedient; however, if they have any plausible doubt about whether their faces are

muddy, they will stay and play, because they are children, after all. Thus, every child, as soon as

he or she can conclude definitively that his$/$her face is muddy, will go home to wash it$.$ But if a

definitive conclusion cannot be made, the child will stay and play. And, of course, pointing out

that someone$$s face is muddy is against their honor system.

There is a traffic light, and children must wait for the walk sign before they go home.

An honest adult named David, who does not know the honor system, comes up to the children,

says $$At least one of you has a muddy face,$$ and leaves. You are too far to see the children$$s faces,

but you know the rules, hear what David is saying, and you can see the children and the traffic

light.

a$)$ Write down the proposition $($with variables A$,$ B$,$ and C$)$ that represents David$$s statement.

b$)$ You observe that Anna does not walk home the first time the walk sign turns on after David$$s

statement. What propositional formula using variables B and C must be true? Explain why.

c$)$ You observe that Bj$$orn and Cathy also do not walk home the first time the walk sign turns on$.$

Think about what can you conclude about A and C$,$ and about A and B$,$ similarly to the previous

part $(2$c from lab$).$ Since now you have three conclusions that must all be true $($two from this part

and one from the previous$),$ combine all three of them into a single propositional formula that you

know must be true.

d$)$ Note that the children can make the same conclusions as you do$,$ because they observe the

same as you do $($even more, in fact$).$ Thus, after the first light cycle, they all know that the formula

from the previous part must be true. The second time the walk sign turns on$,$ you observe that

Anna again does not walk home. What propositional formula using variables B and C must be

true now? Remember that Anna walks home if she is sure her face is muddy, and does not walk

home if she sees that there$$s some way for the formula from the previous part to be true even if

her face is not muddy. Explain your answer.

e$)$ Suppose the second time the walk sign turns on$,$ you observe that in fact no child walks home.

What propositional formula using variables A$,$ B$,$ and C must be true now? Justify your answer

f$)$ Given the observations of the previous part, which child$($ren$)$ will walk home the third time

walk sign turns on and why? Explain your answer.