3 KiB
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.