🚧 Going through 1.3

chapter_1/1_3/investigate.md

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+# Investigate!
+
+Holmes always wears one of the two vests he owns: one tweed and one mint green.
+He always wears either the green vest or red shoes. Whenever he wears a purple
+shirt and the green vest, he chooses to not wear a bow tie. He never wears the
+green vest unless he is also wearing either a purple shirt or red shoes.
+Whenever he wears red shoes, he also wears a purple shirt. Today, Holmes wore a
+bow tie. What else did he wear?
+
+## Try it 1.3.1
+
+Spend a few minutes thinking about the _Investigate!_ question above. Of the six
+statements in the puzzle, only one is atomic. Use this atomic statement and one
+other statement to deduce a new statement about what Holmes might (or might not)
+be wearing. Explain why you think your new statement is true.
+
+**Hint**
+
+The atomic statement is, "Holmes wore a bow tie." Only one of the molecular
+statement has this as one of its atoms.
+
+A:
+
+Let $B$ be "Holmes wears a bow tie." Also, let' $P$ be "Holmes wears a purple
+shirt" and $G$ be "Holmes wears a green vest".
+
+Based off the molecular statement:
+
+"Whenever he wears a purple shirt and the green vest, he chooses not to wear a
+bow tie."
+
+We can write this as:
+
+$$ (P \wedge G) \to \neg B $$
+
+But we know that $B$ is true from the problem statement, which means that Holmes
+is not wearing a purple shirt and the green vest:
+
+$$ \neg (P \wedge G) $$
+
+Let's now write out the other statements. Let $R$ be "Holmes wears the red
+shoes." We know from the problem statement that "He always wears either the
+green vest or red shoes." This is written as:
+
+$$ G \vee R $$
+
+Let $T$ be "Holmes wears the tweed vest." We know from the problem statement
+that "Holmes always wears one of the two vests he owns: one tweed and one mint
+green." This is written as:
+
+$$ T \vee G $$
+
+"He never wears the green vest unless he is also wearing either a purple shirt
+or red shoes.":
+
+$$ G \to (P \vee R) $$
+
+"Whenever he wears red shoes, he also wears a purple shirt."
+
+$$ R \to P $$
+
+This gives us everything we need, let's investigate what we know, and track back
+through the problem to find out what Holmes is wearing.
+
+$$ B \to \neg (P \wedge G) $$
+
+While Holmes could be wearing either the purple shirt or the green vest, he
+cannot wear them together. Let's assume he's wearing the green vest:
+
+$$ G \to (P \vee R) $$
+
+So he can't wear the purple shirt, but he can wear the red shoes.
+
+$$ R \to P $$
+
+Ah, that doesn't work, whenever Holmes wears the red shoes he also wears a
+purple shirt. Therefore Holmes cannot be wearing the green vest.
+
+$$ \neg G $$
+
+So, now we consider "He always wears either the green vest or red shoes."
+
+$$ G \vee R $$
+
+Since we know that Holmes isn't wearing the green vest, therefore he must be
+wearing the red shoes:
+
+$$ R $$
+
+And if he's wearing the red shoes, he is also wearing a purple shirt:
+
+$$ R \to P $$
+
+We also know that Holmes always either wears one of the two vests, the tweed or
+the mint green.
+
+$$ T \vee G $$
+
+Since we know he's not wearing the green vest, he must be wearing the tweed
+vest.
+
+So Holmes is wearing a tweed vest, a bow tie, a purple shirt, and red shoes.