[Math] Variation of a “Who is Lying” Question

problem solving

While reading Problem Solving Strategies – Crossing the River with Dogs and Other Mathematical Adventures by Ken Johnson and Ted Herr, I came across a problem I was not sure how to solve.

One of the five members of a company's board of directors was suspected of sleeping during a board meeting. It was known that only one board member had actually slept, but no one (except the five members) knew who it was. The company vice president questioned the members and they made the following statements:

Davis: The snoozer was either Rawls or Charlton

Rawls: Neither Vongy nor I was asleep.

Charlton: Both Rawls and Davis are lying.

Bobbins: Only one of the Rawls or Davis is telling the truth.

Vongy: Bobbins is a liar.

When the board chairperson (she was not questioned) was consulted, she said that three of the board members always tell the truth and two of them always lie. Who slept in the meeting?

Perhaps I am simply glossing over one facet of the information given that I need to solve it. It seems like a simple problem on its hinge, so I am a bit flustered I am not able to solve it!

If you could give me a run-down of the logic used to solve it, that would be lovely.

Note: this is NOT homework – simply for my own benefit.

Best Answer

If Charlton is telling the truth, that means that both Davis and Rawls are the liars; in particular, Vongy is telling the truth, which would mean that Bobbins is also a liar, contradicting the fact that there are only two liars and there three truth-tellers.

So Charlton is lying. That means that at most one of Davis and Rawls are lying.

If Bobbins is telling the truth, then the second liar must be one of Rawls and Davis, which would again mean that Vongy is telling the truth, making Bobbins a third liar; this is impossible, so Bobbins is lying.

So we now know that the two liars are Charlton and Bobbins, and the remaining three are truth-tellers.

Therefore, the snoozer is either Rawls or Charlton (since Davis is telling the truth), but cannot be Rawls (since Rawls is telling the truth).

So Charlton is the one who fell asleep.

Alternatively: Bobbins and Vongy cannot both be telling the truth, so at least one of them is a liar. In particular, Charlton cannot be telling the truth, since that would give at least three liars. So the two liars are among Charlton, Bobbins, and Vongy, hence both Davis and Rawls are telling the truth; this suffices to establish that Charlton is the one who fell asleep.

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I think the problem is that your sum is not right. It should be $$ \begin{align} \sum_{k=0}^n\left(\frac1{\frac\pi2+\pi(k+1)}+\frac1{\frac\pi2+\pi k}\right) &=\frac2\pi+\frac2\pi\sum_{k=1}^n\frac1{k+\frac12}\\ &\ge\frac2\pi+\frac2\pi\sum_{k=1}^n\frac1{k+1}\\ &=\frac2\pi\sum_{k=1}^n\frac1k \end{align} $$

[Math] What’s the Total Number of Candidates who Applied for the Exam

Your looking at the question innaccurately. Take it slowly and break it down.

In an examination, 5% of the applicants were found ineligible and 85% of the eligible candidates belonged to the general category.

First take the first part

5% of the applicants were found ineligible

What does this tell you? It tells you that 5% were ineligible and 95% were eligible. Let x be the number of total applicants, and e be the number of eligible candidates. We would get the following formula

x = (5/95) e + e

Now we look at the next line

85% of the eligible candidates belonged to the general category.

This is saying 85% of e (not x) belong to the general category. So letting n be the number of people not in the general category we would get

e = n + (85/15)n
e = n * 20/3

We are then given n

 If 4275 eligible candidates belonged to other categories

So we have

n = 4275

Thus we have

e = 4275 * 20/3
e = 28500

And plugging this into our previous equation we get

x = 28500 + (5/95) * 28500
x = 30000

Hope that helps a little more.

Edit Where does 85/15 come from?

Let e = eligible, n = non-generally eligible, and g = generally eligible. So we have

e = n + g

What we are trying to do is convert g to n since we don't know what g is. We do know that:

n = 0.15e
g = 0.85e

Thus we get that

n/0.15 = e
g = (0.85) * (n/0.15)
g = (0.85/0.15) n
g = (85/15) n

And that's where we get the 85/15. Basically think of it as "how much bigger is g than n?". If n is 15% of e and g is 85% of e then g is (85/15) times bigger than n so we get (85/15)n = g.

2nd edit

Why can't we convert (5/95) to (5/100)x?

Because of the following:

x = (5/95) e + e
x = (5/95) e + (95/95) e
x = (100/95) e
e = (95/100) x

You can put it all into one nice equation, but you can't just flop things around without going line by line to see what everything does.

Best Answer

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