Question

Two mathematicians, Tom and Smith are walking down the street.

Tom: I know you have three sons. What are their ages?
Smith: The product of their ages is 36.
Tom: I can’t tell their ages from that.
Smith: Well, the sum of their ages is the same as that address across the street.
Tom: I still can’t tell.
Smith: The eldest is visiting his grandfather today.
Tom: Now I know their ages.

Do you know the age of the kids?

Answer 1

The eldest is 9 years old and the 2 younger ones are 2 years old.

Let’s break it down. The product of their ages is 36. So the possible choices are:

1,1,36 – sum(1,1,36) = 38
1,6,6 – sum(1,6,6) = 13
1,2,18 – sum(1,2,18) = 21
1,3,12 – sum(1,3,12) = 16
1,4,9 – sum(1,4,9) = 14
2,2,9 – sum(2,2,9) = 13
2,3,6 – sum(2,3,6) = 11
3,3,4 – sum(3,3,4) = 10

Six of the sums are unique, so if it were one of those, Tom would have recognised the number across the street that matches and he would know the answer, but he could not figure out the answer. This means there are two or more combinations with the same sum. From the choices above, only two of them are possible now.

1,6,6 – sum(1,6,6) = 13
2,2,9 – sum(2,2,9) = 13

When Tom heard that the eldest is visiting his grandfather, we can eliminate combination 1 since there are two eldest ones. This leaves us with only 1 option left, that is 2, 2 and 9.

Answer 2

» The eldest kid is a 9 yr. old and the other two younger kids are 2 years old each.

=> Product of their ages = 9*2*2 = 36

» The possible choices could be:-
(i) 1, 1, 36 = 1 + 1 + 36 = 38
(ii) 1, 2, 18 = 1 + 2 + 18 = 21
(iii) 1, 3, 12 = 1 + 3 + 12 = 16
(iv) 1, 4, 9 = 1 + 4 + 9 = 14
(v) 1, 6, 6 = 1 + 6 + 6 = 13
(vi) 2, 2, 9 = 2 + 2 + 9 = 13
(vii) 2, 3, 6 = 2 + 3 + 6 = 11
(viii) 3, 3, 4 = 3 + 3 + 4 = 10

=> Out of the above 8 sums, 6 sums are unique in nature. Tom would've been able to recall the matching number across the street, if the number was out of those 6 unique sums.
=> Since Tom couldn't make out the solution to this problem, this implies that two or more combinations exist with the same sum.

» Therefore, just two of them are possible now from the above mentioned choices:-

(a.) 1, 6, 6 = 1 + 6 + 6 = 13

(b.) 2, 2, 9 = 2 + 2 + 9 = 13

» When Tom heard about the eldest kid's visit to his grandfather, Combination (a.) can be eliminted (Reason: There are 2 eldest kids).

» Therefore, we're left with just option (b.) i.e. 2,2,9.
_________________________

Answer --> Ages of the younger kids = 2 years each.
Age of the eldest kid = 9 years.

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