You Be The Detective - 2

Logic Level 1

Another robbery is committed. This time though, Rocky, Bob and Mike (the friends of the trio from You Be The Detective - 1 ) are brought for questioning. Sub-Inspector Milind found that:

$(1)$ If Rocky is guilty and Bob is innocent, then Mike must be guilty.

$(2)$ Mike never works alone.

$(3)$ Rocky and Mike never work together.

$(4)$ No one other than Rocky, Bob and Mike are involved and at least one of them is guilty.

Only you can help Milind now, to find the person who is necessarily guilty.

This problem is the part of my set Is This What You Call Logic?!

Rocky Bob Mike Milind

3 solutions

Caleb Townsend
Feb 12, 2015

From statement $(3),$ we know that Rocky's guilt implies Mike's innocence (and vice versa). But from statement $(1),$ if Rocky is guilty and Bob is innocent, then Mike is guilty; but Rocky's guilt implies Mike's innocence, so we can rewrite statement $(1)$ as $\text{If Rocky is guilty then Bob is guilty.}$ Now let's consider the case that Mike is guilty. From statement $(2),$ we know that if Mike is guilty, then at least one other person is guilty. Combining this with statement $(3),$ we can rewrite it as $\text{If Mike is guilty, then Bob is guilty.}$ The final case to consider is that neither Rocky nor Mike is guilty, which from statement $(4)$ would imply that Bob is guilty. It is a tautology that if Bob is guilty, then Bob is guilty, so in all 3 cases, we have that $\boxed{\text{Bob is guilty.}}$

No way.. it says Bob is innocent.. how come is he theN Guilty

Sunmola Anuoluwapo - 5 years, 8 months ago

John Wong
Mar 3, 2015

if rocky and mike never work together and at least one of them is guilty, Bob is guilty

That makes no sense.

Rocky and Mike never working togheter might mean that either Rocky or Mike might have done it, alone.

Carlos Merino - 5 years, 9 months ago

But the statement says that Mike never does work alone. see the the logic there?

John Wong - 5 years, 8 months ago

But Rocky can, right?

Saya Suka - 4 years, 5 months ago

You made a leap of logic here. Why don't you clarify some things, like why can't Rocky be guilty alone?

Saya Suka - 4 years, 5 months ago

Because if Rocky is guilty, and Bob is innocent (meaning that Rocky did work alone), then Mike is also guilty (so Rocky did not work alone after all), BUT, Rocky and Mike don't work together... And so it means that Bob is the only one that we can be SURE is guilty.

Rafael Fontoura da Silva - 4 years, 3 months ago

I still can't follow your logic.

Saya Suka - 4 months ago

Saya Suka
Jan 11, 2017

Consider statements 2 & 3 first. Mike needs help, but it's not Rocky's. So the culprit(s) could either be.

1) Mike and Bob.
2) Rocky alone.
3) Rocky and Bob.
4) Bob alone.

Now look at statement 1. For "if p, then q" to be true, there are 3 ways for that to happen. Either (p,q) being (true,true) or (false,true) or (false,false). But we can see that p and q cannot both be true, since statement 3 said Mike and Rocky never work together. Thus for the statement to stay true, p must always be false, either Rocky is innocent or Bob is guilty or both. If Rocky is innocent (looking at the 4 possibilities, we'd just ignore #2 & #3), then Bob will have to be guilty, working alone or with Mike. If Bob is guilty (we ignore possiblity #2 among the 4), then he might have done it alone or with either of the two company. Anyway, Bob is guilty.

(1) If Rocky is guilty and Bob is innocent, then Mike must be guilty.

i) Bob is guilty alone
Rocky is guilty and Bob is innocent ==> False from (F×F)
Mike must be guilty ==> False

ii) Bob is guilty along with Mike
Rocky is guilty and Bob is innocent ==> False from (F×F)
Mike must be guilty ==> True

iii) Bob is guilty along with Rocky
Rocky is guilty and Bob is innocent ==> False from (T×F)
Mike must be guilty ==> False

You Be The Detective - 2

3 solutions

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