Maths
posted by Jane .
"How old are your three children?" the maths teachers ask a former student. He is told their ages add to 13, and multiply to give the number on his study door (which they can both see). "i will need to know more", the master days, after a few moments relfection. "the eldest one is learning to play the violin", replies his former pupil. "ah! in that case i can now give you their ages", the master tells him, and does so correctly. How does he know? what are their ages? what is the number of the door?

1. list all the possible ages of the children (you don't need to, cos i've done it for you) 
eg. (1, 1, 11) (1, 2, 10) etc
2. if you list all this, you will find that there are 2 answers that give a solution of 36. these are:
(6, 6, 1) and
(9, 2, 2)
3. the master says he needs to know more, because there are solutions from which he must choose, and in order to get the right answer, he needs one more clue.
4. the former student says that "his eldest son...", meaning that in the 3 sons, there is one that is older than the rest.
5. from the 2 options listed:
(6, 6, 1) and (9, 9, 2), we can see that (9, 2, 2) is obviously the answer, because the 9 year old is older than the rset, whereas in the other case, there are two sons with the same age of 6.
6. their ages are 9, 2, and 2, and the number on the door is 36. 
The mathematician will only beed to know more if there are two products that are the same (36), 1,6,6, or 2,2,9. In the question it mentions an 'eldest' son, meaning that there is only 1 eldest son (not twins). Therefore their ages are: 2,2,9 and the office number is 36.