91av

Answer: 1, 8 and 9                                    

The winner is Dieter Stahel of Brugg, Switzerland.

Worked solution            

The possible triples of ages with product 72 is given below (with the sum of the three in brackets):

1/1/72 (74); 1/2/36 (39); 1/3/24 (28); 1/4/18 (23); 1/6/12 (19); 1/8/9 (18); 2/2/18 (22); 2/3/12 (17); 2/4/9 (15); 2/6/6 (14); 3/3/8 (14); 3/4/6 (13)

Case 1

Houses are all odd or all even.             

If you are told at house N that the total is N-2 or N+2 but that gives more than one triple in the above list, then N must be one of

12, 15, 16, 17, 20, 21                               

But the pattern of house-numbers of the first four responses are

X, X+2, X+4, …, Y                                    

´Ç°ùÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌýÌý

X, X-2, X-4, … , Y                                    

and that is impossible in this case.            

Case 2            

Houses are numbered consecutively.             

If you are told at house N that the total is N-1 or N+1 but that gives more than one triple in the above list, then N must be one of

13, 14, 15, 16, 18                                     

But the pattern of house-numbers of the first four responses are

X, X+2, X+4, …, Y                                     

or                                     

X, X-2, X-4, … , Y                                     

and so Y must be 18.                               

Hence the house called out in the last paragraph is 19 and the sum is 18 or 20, uniquely determining the triple 1/8/9.