Solution: To Tell the Truth Math Puzzle
We had several people suggest solutions to this puzzle.
I think this is a solution: If you can see the people you are interviewing with, start with
Question 1: Are you a man?
This question, or a similar question with a known answer, will show whether that person tells the truth or not.
If you get a true answer, then continue with
Question 2a: Do all the people in your company always tell the truth?
If the answer is “yes,” this is the company where people always tell the truth. If the answer is “no,” this is the half-and-half company, because at least the person you’re speaking with told the truth, so they don’t always lie.
Now back to Question 1. If that answer was a lie, then continue with
Question 2b: Does anyone in your company tell the truth?
If the answer is “yes,” then this is the company of liars. Since they have to lie all the time, the real answer is “no.” If the answer is “no,” then it’s the half-and-half company.
If you interview people from each company, you’ll be able to sort them out this way.
I found the solution to a similar problem: Of three men, one always tells the truth, one always tells lies, and one answers "yes" or "no" randomly. Each man knows which one each of the others is. You may ask three yes/no questions, each of which may only be answered by one of the three men, after which you must be able to identify which man is which. How can you do it?
There are six possible scenarios. Let's call the first man A, the second man B, and the third man C. The six scenarios, then, are:
|
Scenario
|
A
|
B
|
C
|
|
I
|
Truth teller
|
Liar
|
Random
|
|
II
|
Truth teller
|
Random
|
Liar
|
|
III
|
Liar
|
Truth teller
|
Random
|
|
V
|
Liar
|
Random
|
Truth teller
|
|
IV
|
Random
|
Truth teller
|
Liar
|
|
VI
|
Random
|
Liar
|
Truth teller
|
Follow these steps to determine which possibility listed above is correct:
- Ask A, "Is B more likely to tell the truth than C?"
If yes, go to step 2.
If no, go to step 5.
- Ask C, "Are you the random man?"
If yes, go to step 3.
If no, go to step 4.
- Ask C, "Is A the truth teller?"
If yes, then scenario V is the case.
If no, then scenario II is the case.
- Ask C, "Is A the liar?"
If yes, then scenario IV is the case.
If no, then scenario VI is the case.
- Ask B, "Are you the random man?"
If yes, go to step 6.
If no, go to step 7.
- Ask B, "Is A the truth teller?"
If yes, then scenario VI is the case.
If no, then scenario I is the case.
- Ask B, "Is A the liar?"
If yes, then scenario III is the case.
If no, then scenario V is the case.
The trick is to ask a "double question," such as "If I were to ask you if you work for Company X, would you say Yes?" By so doing, you will get the correct answers to every question you ask without having to correlate answers to different questions.
- If the person always tells the truth, you will always get the correct answer.
- If the person always lies, this person will (a) lie if he/she works for Company X and (b) lie about the answer. Double negatives will result in a positive.
This principle works on the individual who tells the truth or lies (T or F.) It is independent of the number of corporations or tribes (groups of T's and F's.) Thus, the complexity of the third tribe is simply trying to throw people off.
