riddle of the day

Two mathematicians, a visible route number, and a single denial that reveals one passenger's age.

The Conway Bus

Master
Numerical territory

Riddle statement

Two mathematicians are riding a bus, and both can see the route number.

One of them says:

—I have at least two children. Their ages are positive integers and may be repeated. The sum of their ages is this route number; their product is my age.

The other replies:

—If I also knew your age and how many children you have, I could determine their ages exactly.

—No, you could not.

After thinking for a moment, the second mathematician concludes:

—Then I know how old you are.

What is the route number, and what age has been deduced?

Show solution

Solution

Answer: the route number is 12, and the father is 48 years old.

The father's denial means that even knowing the sum of the ages, their product, and the number of children would still leave at least two possible age lists.

For every route number \(s\), we must therefore find the products arising from two distinct lists with:

  • the same sum \(s\);
  • the same number of terms;
  • the same product.

The second mathematician can determine the age only when exactly one ambiguous product exists for the visible sum.

The check is exhaustive: generate all nondecreasing lists of at least two positive integers with sum \(s\), then group them by their length and product. The first results are:

  • for \(2\le s\le11\), there is no ambiguous product;
  • for \(s=12\), the only ambiguous product is \(48\);
  • for \(s=13\), both \(36\) and \(48\) are ambiguous.

For route 12, the two responsible lists are

$ 1,3,4,4 $

and

$ 2,2,2,6. $

Both have four terms and satisfy

$ 1+3+4+4=2+2+2+6=12, $
$ 1\cdot3\cdot4\cdot4 = 2\cdot2\cdot2\cdot6 = 48. $

Thus, even knowing that the father has four children and is 48 years old would not determine their ages. Since 48 is the only ambiguous product for sum 12, the denial reveals his age.

Why can the route not be larger? At sum 13 there are two different ambiguities:

$ 1,6,6 \quad\hbox{and}\quad 2,2,9, $

with product 36, and

$ 1,1,3,4,4 \quad\hbox{and}\quad 1,2,2,2,6, $

with product 48.

Adding the same number of one-year-old children to both lists in a pair increases their sums and lengths equally while leaving their product unchanged. The ambiguities 36 and 48 therefore survive for every sum greater than 13.

For a route below 12 there would be no ambiguity; for a larger route there would be at least two possible ages. The only route consistent with the conversation is 12, and the deduced age is 48.

The essential idea is to find the unique sum that permits ambiguity for exactly one product.