The bus of Conway

1. The answer “You couldn't” speaks of a very specific ambiguity: same sum, same product and same number of children.
2. Find a sum for which this ambiguity exists, but leave only one possible age.
3. With sum 12 two different distributions of four children appear with the same product.

Answer: the bus line is 12. The second mathematician knows the number of the line, so he knows the sum of the ages. The important phrase is: > “If I knew your age and how many children you had, I could deduce their ages.” The father answers: > “You couldn't.” This means that, for the sum of that line, there are at least two possible distributions of ages with: - the same number of children;

• the same sum;
• the same product. Because if the second mathematician knew the age of the father, he would know the product; and if he also knew the number of children, he would only fail if there were still two indistinguishable distributions. With sum 12, this happens: $$

1,3,4,4

$$ and $$

2,2,2,6.

$$ Both groups have four children. Both add up to 12: $$

1+3+4+4=12,

$$ $$

2+2+2+6=12.

$$ And both have product 48: $$

1\cdot3\cdot4\cdot4=48,

$$ $$

2\cdot2\cdot2\cdot6=48.

$$ Therefore, if the line is 12 and the father is 48 years old and has four children, not even knowing the age and number of children would be enough to reconstruct the ages. Now it remains to be seen why this allows age to be deduced. For sums less than 12 no ambiguity of this type appears. That is, if the line were less than 12, the answer “You couldn't” could not be given. With sum 12 a single ambiguous age appears: 48. And for any sum greater than 12, at least two ambiguous ages appear. In fact, it is enough to add ones of age 1 to the two previous examples: $1,3,4,4$ and $2,2,2,6$ maintain product 48 when adding the same ones to both distributions. Furthermore, from sum 13 another ambiguity also appears: $$

1,6,6

$$ and $$

2,2,9,

$$ that have the same sum 13, the same number of children and the same product: $$

1+6+6=13,

$$ $$

2+2+9=13,

$$ $$

1\cdot6\cdot6=36,

$$ $$

2\cdot2\cdot9=36.
$$ If the sum is greater than 13, the same ones are added to both distributions and the ambiguity is preserved. Thus, for every line greater than 12 there would be at least two possible ages, 48 ​​and 36, compatible with the phrase “You couldn't”. Therefore, the only line in which the father's refusal leaves a certain age is: $$12.$$ Along these lines, the age that the second mathematician deduces is 48.