Two friends, Alex and Bob, go to a bookshop, together with their sons Peter and Tim. All four of them buy some books. Each book costs a whole amount of shillings.

When they leave the shop, they notice that both fathers have spent 21 shillings more than their respective sons. Moreover, each of them paid per book the same amount of shillings as the number of books that he bought.

The difference between the number of books that Alex bought and that Peter bought is five.

Who is the father of Tim?

(In reply to

answer by K Sengupta)

By the given conditions, we have:

p^2 - q^2 = 21, where p and q denote the respective number of books bought by a father and his son. It may be observed that although the pair (p, q) may differ for the two families, the above equation is satisfied for both.

Now, p^2 - q^2 = 21

or, r(p+q) = 21, where p-q= r(say)

or, p+q = 21/r

Solving for (p+q, p-q) = (r, 21/r), we have:

(p, q) = (r + 21/r)/2), (r - 21/r)/2)

Thus, r > 21/r, so that: r^2 > 21, so r>= 4, so that r divides 21

The only possible positive integer value for r occurs at r = 7, 21

giving 21/r = 3, 1, so that: (p, q) = (5, 2), (11, 10)

Thus, respective number of books purchased by one father-son duo are 5 and 2, while the respective number of books purchased by the other father-son duo are 11 and 10.

The absolute difference between the number of books purchsed by all possible pairs of two given individuals amongst the four individuals are thus: (3, 6, 5,1, 9, 8), so that the only way this difference can equal 5, is for the individual having bought the greater number of books corresponding to the father of one family, while the individual having bought the lesser number of books being the son of the other family.

Accordingly, by the given conditions Alex is the father of one family, while Peter is the son in the other family.

Since Alex is not the father of Peter, it follows that he must be the father of Tim.