A new addition to the family

Logic Level 4

One joyous morning, Mrs. Tan gave birth to her fifth child, Eric, in Singapore General Hospital. Later in the day, the family of seven bumped into her old friend who was surprised to see a newborn baby. The friend asked how old her four older children were.

Mr. Tan said, "The sum of the ages of our four older children is $x$ (an integer less than 25). The product is 216."

The friend said, "This isn't helpful. Tell me more."

Mr. Tan replied, "Our oldest child's name is Charles."

The friend said, "Still not enough information."

Mr. Tan answered, "Including Eric, our only daughter, Brenda, is now our middle child."

Mrs. Tan's friend answered, "Aha! Now I know the answer!"

Assuming Mrs. Tan's friend is perfectly logical, how old are the four older children? Give your answer as the concatenation of the four ages in descending order. For example, if the four older children are 1, 2, 3 and 4 years old, your answer is 4321.

Note : Mrs. Tan's friend knows what is $x$ but we as the readers do not.

The answer is 9641.

2 solutions

Noel Lo
Jul 4, 2016

The following image shows the possible combinations and their corresponding sums:

That Mrs Tan's friend was still not sure even after being told the sum and product we eliminate those combination's with a unique sum. This leaves us with the combinations summing to 17, 19 and 20:

We see that Mrs Tan's friend still could not tell even after Mr Tan made reference to the oldest child. So the age sum cannot be 17 or 19 as for each sum, there is one combination with a unique oldest age and one without. Hence the age sum must be 20 where each combinations has a unique oldest age. But we can narrow down to 9641 and rule out 12, 3, 3, 2 as Brenda is the middle child so there must be five children of distinct ages. In 12, 3, 3, 2 we only have four distinct ages including Eric so there is no middle child.

just to complete, there is one more combo... 3, 3, 4, 6 with sum of 16

Satyen Nabar - 4 years, 11 months ago

Snehal Shekatkar
Jul 26, 2016

Since Mrs. Tan says that the product of the ages is $216$ , we have only following combinations of ages for which the sum of ages is less than $25$ :

$2 \times 2 \times 6 \times 9$

$2 \times 2 \times 3 \times 18$

$3 \times 3 \times 4 \times 6$

$1 \times 4 \times 6 \times 9$

However, as Mrs. Tan says that there is a middle child, we can't have two people of the same age and hence top three combinations must be wrong. This leaves us with the last one and hence the answer is $\boxed{9641}$

A new addition to the family

The answer is 9641.

2 solutions

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