Ben has three daughters. Paul, a neighbour, asks how old they are. Ben replies as following.

– The product of their age is 36.

– The sum of their ages is equal to my house number.

Upon hearing this, Paul is still unable to determine the 3 daughter’s age.

Which information would help Paul conclusively determine the 3 daughter’s age?

a. Two of them are twins

b. Neither of them are twins

c. The eldest is blonde

d. The 3 are below 10 years

Right answer is: c) The eldest is blonde

If the product of the three ages is 36 then there are 8 possible ways in which that can happen.

1) 36=36x1x1, the sum is 38

2) 36=18x2x1, the sum is 21

3) 36=12x3x1, the sum is 16

4) 36=9x4x1, the sum is 14

5) 36=9x2x2, the sum is 13

6) 36=6x6x1, the sum is 13

7) 36=6x3x2, the sum is 11

8) 36=4x3x3, the sum is 10

Paul is a neighbour, hence he knows the house number. If after taking account of both the product and the sum, Paul is still unable to work out the ages, then it means that there are several possible answers. That would happen when both sums are 13. Therefore the ages are either (9,2,2) or (6,6,1). Answers b) and d) are not possible, answer a) is not useful, answer c) is determinant since implies that there is one only eldest daughter. The ages are then (9,2,2).

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