# The age of three ladies

Three ladies A, B and C have a discussion, where each lady says the truth twice and lies once.

1. A: B is two years older than me
2. B: C is 38 years old
3. C: A is older than me
4. B: The age difference between C and me is three years
5. C: A is 36 years old
6. A: I'm 35 years old
7. B: At least one of A or C is younger then me
8. A: I'm one year older than C
9. C: A is three years younger than B

What is the age of each of the three ladies?

A says:

• $$B = A + 2$$
• $$A = 35$$
• $$A = C + 1$$

B says:

• $$C = 38$$
• $$|B - C| = 3$$
• $$A < B$$ or $$C < B$$

C says:

• $$A > C$$
• $$A = 36$$
• $$B = A + 3$$

Notice that

A and C have two pairs of conficting statements. This means that they are both correct on one and wrong on the other. And their remaining statements are both true.
Thus $$A = C + 1$$ and $$A > C$$ are true. Also, $$A = 35$$ or $$36$$.

This means that

$$C = 34$$ or $$35$$, making the statement $$C = 38$$ false.
Therefore the other two statements of B are true.

Now we see that

$$|B - C| = 3$$ means that $$B, C$$ have different parity.
But we also know $$A = C + 1$$, hence $$A, B$$ have the same parity.

This forces

$$B = A + 3$$ to be false, thus $$A = 36$$, $$C = 35$$ and $$B = A + 2 = 38$$.

• just beat me to it! well done Dec 6, 2021 at 19:56
• One problem with your logic - "both correct on one and wrong on the other" is not true. The two conflicting statements could have both been lies. Dec 6, 2021 at 20:01
• @RobWatts In each pair of conflicting statements, at least one is false. There are two pairs of conflicting statements between A and C, thus at least two of these statements are false. But A and C have only two false statements in total, one for each. Therefore there is no other false statement - in particular, in each conflicting pair, exactly one statement is false. Dec 6, 2021 at 20:19
• I see what I missed. "A and C have two conficting statements" would be better as "A and C have two pairs of conflicting statements." Dec 6, 2021 at 20:48
• @RobWatts Ah, I see. Thanks for pointing out. I updated the answer to make this clear. Dec 6, 2021 at 20:55