Two friends Jack and Bill discuss the ages of Bills three children; the product of their ages is 72 and the sum of their ages equals Jacks birth date, and using the extra clue that there is a unique eldest child who has just started piano lessons, find the ages of the three children.

Introduction / Context:
This well known puzzle about the ages of Bills three children combines number factorization, logical reasoning, and the use of indirect information. The story mentions the product of the ages, the sum matching a birth date, and an extra clue involving the eldest child. Such questions appear in logical reasoning and arithmetic reasoning sections to test systematic thinking rather than pure calculation speed.

Given Data / Assumptions:
- Bill has exactly three children.
- The product of their ages is 72.
- The sum of their ages equals Jack's birth date, which must be a possible calendar day number between 1 and 31 inclusive.
- Even after knowing the product and the sum, Jack initially remains unsure about the ages.
- The final clue that there is a unique eldest child who just started piano lessons allows Jack to determine the correct ages.

Concept / Approach:
The main technique is to list all possible triples of positive integer ages whose product is 72. For each triple, we compute the sum and interpret it as a possible birth date. The statement that Jack still does not know the ages after hearing that the sum equals his birth date implies that more than one triple corresponds to the same sum, otherwise he would have identified the ages immediately. Therefore we look for sums that appear more than once among the candidate triples. Finally, the information that there is a unique eldest child eliminates any triple where the two oldest ages are equal, since that would imply joint eldest children rather than a single eldest child.

Step-by-Step Solution:
Step 1: List positive integer triples whose product is 72, such as (1, 1, 72), (1, 2, 36), (1, 3, 24), (1, 4, 18), (1, 6, 12), (1, 8, 9), (2, 2, 18), (2, 3, 12), (2, 4, 9), (2, 6, 6), (3, 3, 8), and (3, 4, 6). Step 2: Compute sums: examples include 1 + 6 + 12 = 19, 1 + 8 + 9 = 18, 2 + 6 + 6 = 14, 3 + 3 + 8 = 14, and so on. Step 3: Identify sums that repeat. The sum 14 occurs for two different triples: (2, 6, 6) and (3, 3, 8). This means that if Jack's birth date is 14, he would still be unsure which triple is correct after hearing the product and the sum. Step 4: The story says Jack remains uncertain until Bill adds that his eldest child has started piano lessons. This implies there is a unique eldest child. Step 5: In the triple (2, 6, 6) there are two children aged 6, so there is no single eldest child. In (3, 3, 8) the child aged 8 is uniquely oldest. Therefore (3, 3, 8) is the only triple consistent with all clues.

Verification / Alternative check:
We can quickly verify that 3 * 3 * 8 = 72 and that the sum 3 + 3 + 8 = 14, which is indeed a valid day number for a birth date. Checking the rejected triple (2, 6, 6) also gives a product of 72 and sum 14, confirming why Jack remained uncertain before the final clue. The last statement about the eldest child is precisely what disambiguates the two possibilities. No other triple offers this exact combination of repeated sum, matching product, and existence of a unique eldest child.

Why Other Options Are Wrong:
Option 2, 6, 6: This triple has a correct product and sum, but it has two eldest children tied at age 6, so there is no unique eldest child.
Option 1, 6, 12: The product is 72 but the sum is 19, which would have uniquely identified the ages immediately, contradicting the story of Jack's temporary confusion.
Option 1, 8, 9: Here the product is 72 and the sum is 18, again unique as a sum, so Jack would not have remained unsure after hearing the sum.
Option 2, 3, 12: This also has product 72 but a sum of 17, which is a distinct birth date and gives no ambiguity.

Common Pitfalls:
Many learners focus only on the original product and ignore the narrative clue about Jack's uncertainty. Others may stop after finding any triple that multiplies to 72 rather than systematically listing all possibilities. Another common error is forgetting that a calendar birth date must be between 1 and 31, which restricts the possible sums. A final pitfall is misinterpreting the phrase eldest child and not realizing that any triple with two equal largest ages is ruled out.

Final Answer:
The ages of Bills three children are 3 years, 3 years, and 8 years.