The Ages of Mathematician´s sons Two mathematicians meet and talk:
"Do you have a son?" asked the first mathematician.
"Yes I actually have three sons, and none of them are twins." answered the second mathematician.
"How old are they?" asked the first mathematician.
"The product of their age is equal to the month number at this moment." answered the second mathematician.
"It is not sufficient!" said the first mathematician.
"True, if you sum their ages next year it will again be equal to the month number at this moment." said the second mathematician.
How old are his sons? (I was not able to evaluate this mathematically!) 

Let $A_1,A_2,A_3$ be the ages of the sons respectively. Observe that if the month is $1,2,3,4,5,7,9,11$ then there are no solutions using the fact that there are no twins. If the month is $6,8$ or $10$ then there is a unique solutions so just by the first information it would be possible to determine the ages. Hence the month must be December.
$12$ has two decompositions : $(1,2,6),(1,3,4)$ and the sum of their ages next year is $12$ in the former case and $11$ in the latter case.
Thus the solution is $(1,2,6)$.
