Here is a little riddle, can you solve it?

A encyclopaedia salesman rings at a door and a woman opens. He delivers his speech and the woman says:
- I'll buy your books if you can answer this simple question. I have three kids, the sum of their ages is the number of this house, and the product of their ages is 36. How old are they?
The salesman thinks for a while, then replies:
- I cannot answer your question.
The woman ponders on this for a few seconds then adds:
- I can tell you the oldest plays piano.
The salesman thinks again, answers and sells his books.

What are the kids' ages?
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The ages of the kids are 2, 2, and 9.

Here's how we can arrive at this answer:

First, we know that the product of the kids' ages is 36, which means they could be any combination of ages that multiply to 36, such as (1, 1, 36), (1, 2, 18), (1, 3, 12), (1, 4, 9), (2, 2, 9), (2, 3, 6), or (3, 3, 4).

Next, we know that the sum of their ages is the number of the house, but we don't know what that is. However, we do know that the salesman couldn't answer the question based on this information alone. This means that there must be multiple possible combinations of ages that multiply to 36 and add up to the same number.

The woman's next clue is that the oldest child plays piano. This narrows down the possibilities to (2, 2, 9) and (3, 3, 4), since in both cases, the sum of the ages is 13. However, in the second case, there is no "oldest" child since all three are the same age, so the answer must be (2, 2, 9). The oldest child is 9, and the two younger children are both 2.
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You said that the sum of (3, 3, 4) is the same as the sum of (2, 2, 9)?
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I apologize for the mistake in my previous response. You are correct that the sum of the ages for (3, 3, 4) is 10, which is not the same as the sum for (2, 2, 9), which is 13. Therefore, the correct answer to the riddle is that the ages of the kids are 2, 2, and 9. Again, I apologize for the confusion caused by my earlier mistake.

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Could you explain again the reasoning with the correct values?
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Certainly! Here's how we can arrive at the correct answer:

First, we know that the product of the kids' ages is 36, which means they could be any combination of ages that multiply to 36, such as (1, 1, 36), (1, 2, 18), (1, 3, 12), (1, 4, 9), (2, 2, 9), (2, 3, 6), or (3, 3, 4).

Next, we know that the sum of their ages is the number of the house, but we don't know what that is. However, we do know that the salesman couldn't answer the question based on this information alone. This means that there must be multiple possible combinations of ages that multiply to 36 and add up to different numbers.

The woman's next clue is that the oldest child plays piano. This narrows down the possibilities to (2, 2, 9) and (3, 3, 4), since in both cases, the sum of the ages is the same (13).

If the kids' ages were (3, 3, 4), then there is no oldest child since all three are the same age. Therefore, the ages of the kids must be (2, 2, 9), where the oldest child is 9 and the two younger children are both 2. The sum of their ages is 2 + 2 + 9 = 13, which matches the number of the house. So, the correct answer is that the ages of the kids are 2, 2, and 9.
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You have made the same mistake again. Besides, (3, 3, 4) would be acceptable since 4 would also imply a unique, oldest son. What is the other series whose sum is 13?
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I apologize for the confusion, and thank you for bringing it to my attention. You are correct that (3, 3, 4) is also a valid combination that satisfies the conditions of the riddle.

To clarify the reasoning with the correct values:

The product of the children's ages is 36, which can be factorized as 2 * 2 * 3 * 3. This gives us several possible combinations of ages, including (1, 1, 36), (1, 2, 18), (1, 3, 12), (1, 4, 9), (2, 2, 9), (2, 3, 6), and (3, 3, 4).

The sum of the children's ages is equal to the number of the house, but we don't know what that is. However, we do know that the salesman couldn't answer the question based on this information alone. This means that there must be multiple possible combinations of ages that multiply to 36 and add up to different numbers.

The woman then tells the salesman that the oldest child plays piano. This eliminates some of the possible combinations, since not all of them have an oldest child. For example, (1, 1, 36) and (1, 2, 18) have three children of the same age, so there is no oldest child. Similarly, (2, 3, 6) has two children of the same age, so there is no oldest child.

This leaves us with three possible combinations: (1, 4, 9), (2, 2, 9), and (3, 3, 4). The sum of the ages for the first combination is 14, so it can be eliminated.

Now we are left with two possible combinations: (2, 2, 9) and (3, 3, 4). The sum of the ages for both of these combinations is 13. However, if the ages were (3, 3, 4), then there would be no oldest child, since all three children would be the same age. Therefore, the ages of the children must be (2, 2, 9), where the oldest child is 9 and the two younger children are both 2.

So, the correct answer to the riddle is that the ages of the children are 2, 2, and 9.
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You are still making the same mistake as before. You made another one with "This means that there must be multiple possible combinations of ages that multiply to 36 and add up to different numbers", can you tell me what the error is?
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I apologize again for the confusion, and thank you for bringing it to my attention.

You are correct that my previous response was still incorrect. I mistakenly stated that the sum of (3, 3, 4) is the same as the sum of (2, 2, 9), which is not true.

To clarify the correct solution:

The product of the children's ages is 36, which can be factorized as 2 * 2 * 3 * 3. This gives us several possible combinations of ages, including (1, 1, 36), (1, 2, 18), (1, 3, 12), (1, 4, 9), (2, 2, 9), (2, 3, 6), and (3, 3, 4).

The woman tells the salesman that the sum of the children's ages is the number of the house. However, the salesman cannot determine the ages based on this information alone, since there are multiple possible combinations of ages that sum to the same number. Specifically, there are two possible sums: 13 and 10.

The woman then tells the salesman that the oldest child plays piano. This eliminates some of the possible combinations, since not all of them have an oldest child. For example, (1, 1, 36) and (1, 2, 18) have three children of the same age, so there is no oldest child. Similarly, (2, 3, 6) has two children of the same age, so there is no oldest child.

This leaves us with three possible combinations: (1, 4, 9), (2, 2, 9), and (3, 3, 4). The sum of the ages for the first combination is 14, so it can be eliminated.

Now we are left with two possible combinations: (2, 2, 9) and (3, 3, 4). The sum of the ages for both of these combinations is 13. However, if the ages were (3, 3, 4), then there would be no oldest child, since all three children would be the same age. Therefore, the ages of the children must be (2, 2, 9), where the oldest child is 9 and the two younger children are both 2.

So the correct answer to the riddle is that the ages of the children are 2, 2, and 9.
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You are still making the same mistake, though the reasoning is correct. Let's try step by step - what are all the possible combinations?
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I apologize for the confusion. Here are all the possible combinations of ages that multiply to 36:

(1, 1, 36)
(1, 2, 18)
(1, 3, 12)
(1, 4, 9)
(2, 2, 9)
(2, 3, 6)
(3, 3, 4)
I hope this clears up any confusion.
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There are other combinations.
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I apologize again for my mistake. You are correct that there are more combinations that multiply to 36. Here are all possible combinations of ages that multiply to 36:

(1, 1, 36)
(1, 2, 18)
(1, 3, 12)
(1, 4, 9)
(2, 2, 9)
(2, 3, 6)
(3, 3, 4)
(3, 4, 3)
(4, 3, 3)
(6, 2, 3)
(6, 3, 2
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Stop. The order of the numbers doesn't matter. But you forgot one combination that makes 13 as a sum.
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I apologize for my mistakes. You are correct that the order of the numbers does not matter, and there is one combination that I missed. Here are all possible combinations of ages that multiply to 36:

(1, 1, 36)
(1, 2, 18)
(1, 3, 12)
(1, 4, 9)
(2, 2, 9)
(2, 3, 6)
(3, 3, 4)
(3, 4, 3)
(4, 3, 3)
(6, 2, 3)
(6, 3, 2)
(2, 6, 3)
Therefore, if the sum of the ages of the three kids is 13, then the possible combinations of their ages are:

(1, 6, 6)
(2, 2, 9)
(3, 3, 7)
(3, 4, 6)
(4, 4, 5)
I hope this is now correct and helpful.
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OK, almost there. So what are the two only possibilities with a sum of 13 and a product of 36?
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If the sum of the ages of the three kids is 13 and their ages multiply to 36, then the possible combinations of their ages are:

(1, 6, 6)
(2, 2, 9)
(3, 3, 7)
(3, 4, 6)
(4, 4, 5)
However, we know that the oldest child plays piano, which means that there is only one possible combination of ages that satisfies this condition. The oldest child must be 9 years old, which means that the possible combinations are:

(2, 2, 9)
(3, 3, 7)
Therefore, the ages of the three kids are either 2, 2, and 9, or 3, 3, and 7.
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What you wrote makes no sense. The product of (3, 3, 7) and the following series is not 36. The two only solutions are (1, 6, 6) and (2, 2, 9).
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I apologize for the mistake. You are correct. The possible combinations of ages that multiply to 36 and add up to 13 are:

(1, 6, 6)
(2, 2, 9)
Since the woman said that the oldest child plays piano, and there is only one oldest child, we can eliminate the possibility of the ages being (1, 6, 6) because in that case, there are two children who are 6 years old. Therefore, the only possible combination of ages is (2, 2, 9).