how old are your children

Question

Andrew and Bert met on the street and had the following conversation: A: How old are your three children? B: The product of their ages is 36. A: That’s not enough information for me to know their ages. B: The sum of their ages is your house number. A: That’s still not quite enough information. B: The oldest child plays the piano. A: Now I know! Assume that the ages are whole numbers and that twins have the same age. How old are the children?

Answer 1

Andrew and Bert met on the street and had the following conversation:

A: How old are your three children? B: The product of their ages is 36.

A: That’s not enough information for me to know their ages. B: The sum of their ages is your house number.

A: That’s still not quite enough information. B: The oldest child plays the piano. A: Now I know! Assume that the ages are whole numbers and that twins have the same age. How old are the children?

 

This problem uses factors of numbers to supply answers to the puzzle.

 

“A: How old are your three children? B: The product of their ages is 36.”

Important math information:  3 values; product is 36

This means that we start with all the factors of 36:  1, 2, 3, 4, 6, 9, 12, 18, 36.

Then, we select all combinations of 3 of those and determine whether the product is 36.  This could be multiple steps, but here are the products in order of sums (looking ahead to next Q&A):

   10       3  3  4

   11       2  3  6

   13       1  6  6

   13       2  2  9

   14       1  4  9

   16       1  3  12

   21       1  2  18

   38       1  1  36

   [Full disclosure: computer program -- For A=1 TO 36; For B=A TO 36; FOR C=B TO 36;  IF(A+B+C)=36 THEN

      PRINT((A+B+C),A,B,C); NEXT C; NEXT B; NEXT A;

    sort the results into ascending sum order]

   

A: That’s not enough information for me to know their ages. B: The sum of their ages is your house number.

There are several sets of three values whose product is 36, so “That’s not enough information.”  “The sum … is your house number” means that Bert can immediately identify the three numbers whose sum equals his house number – unless, there is a duplicate (which is why we sorted the values by sum).  The only duplicate sum is 13.

 

A: That’s still not quite enough information. B: The oldest child plays the piano.

Since there are two possibilities for three values (two of which are “twins” of the same age), “That’s still not quite enough information.”  “The oldest…” means that this child is not a twin, so the ages must be 2, 2, 9.

 

A:  “Now I know!”

This is “the process of elimination.”