Im Possible

From the very first statement, we gather that the product of daughter's age is 36

We have:

• {4,3,3}
• {6,6,1}
• {2,2,9}

Now the mother says, "I'd tell you the sum of their ages, but you 'd still be stumped." Which means, the sum of the ages wouldn't be a viable option to deduce the answer. This implies that the sum of ages should be the same, since that is the only way the clue could be useless. Hence,

Now we have

• {6,6,1}
• {2,2,9}

Now the daughter mentions her eldest daughter, this would mean that the eldest daughter is not a twin, and there is only one eldest daughter.

Therefore, we wound up with:

• {2,2,9}

As the problem stated, the product of the ages is 36 so answer should be (2, 2, 9) or (1, 6, 6). From the problem statement, the mother claimed that the oldest daughter Annie likes dogs, so answer choice (1, 6, 6) should be eliminated, as this indicates there will be two eldest daughters.

We rule out 9, 3, 1 from the options as the product is 27, not 36. Since the sum of ages is not enough, we find that we need 2 sets where both the sum and product of the 3 numbers are the same. 4, 3, 3 is not feasible as there is no other set of 3 numbers whose product = 36 and sum=10. 9, 2, 2 and 6, 6, 1 would be 2 possible sets but bear in mind that the two oldest cannot be of the same age (OLDEST daughter) so we go for 9, 2, 2.