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On New Year's Eve, a census taker gathering information calls a woman and asks specific questions about her family and their (integer) ages.

She replies, "I don't like to give out specifics, but it's just me and my four or five children, all biological, here. I won't give you our exact ages, but I can tell you that the product of our ages is 2,310 and the sum of our ages is odd."

A few moments of silence go by as the census taker runs some calculations.

"Can you give me any more information, possibly?" said the census taker.

She thinks for a minute and then says, "The sum of my children's current ages is at least 18."

The census taker does some calculation and claims not to have enough information.

"One more quick question, if I may," he says.

"I think I've answered quite enough," she says as she starts to hang up.

He quickly asks, "Are any of your children twins, triplets, or quads?"

"No," she answers as she ends the call.

The census taker checked his calculations and recorded the woman's age, along with the number of her children and their ages.

What did he record?

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  • $\begingroup$ And does no twins/triplets also exclude quads, quints etc? $\endgroup$
    – fljx
    Jun 8, 2022 at 7:48
  • $\begingroup$ @JLee With quads allowed there are multiple possible solutions. $\endgroup$
    – fljx
    Jun 8, 2022 at 13:56

4 Answers 4

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The prime factors of 2310 are 2,3,5,7 and 11, once each.

In addition to that,

the product may include one or two ones: it's entirely possible to have two children born, say, 11 months apart, so they are the same age even though they aren't twins.

So, distributing the prime factors to the persons, we find only one solution:

Mother 35, kids 11,3,2,1,1

This distribution of the primes is (with some reservations) unique: one of the children must be 11; otherwise the sum of the kids' ages cannot reach 18 (the mother's age must use up one of the remaining primes in addition to the 11, and even choosing the smallest one, we get 7+5+3+1+1 < 18). And with one of the kids being 11, we really don't want to make the mother as young as 21, which is the next biggest option. (We cannot afford to spend three primes on the mother's age, since then we'd need to have 2 one-year-olds to reach 4 children, and the sum of all ages cannot be odd anymore.)

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    $\begingroup$ Except your final solution is even, not odd. However, you probably have the correct answer lower down with the mother being 21, married to an older man who already had kids. A 40 yo remarrying an 18 yo is not unheard of cough- my father- cough. That being so, it's no wonder she doesn't want to share her personal data... $\endgroup$
    – mkinson
    Jun 8, 2022 at 11:27
  • $\begingroup$ @JLee that would require the lateral-thinking tag. $\endgroup$
    – Florian F
    Jun 8, 2022 at 11:42
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    $\begingroup$ @mkinson The oddness requirement concerns the combined age of the entire family, which comes up to 53. $\endgroup$
    – Bass
    Jun 8, 2022 at 11:54
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    $\begingroup$ Great job. You nailed it. The "no twins or triplets" thing was intended to confuse, and boy did it ever confuse a lot of people, but you saw right through it, as well as the "the sum of our ages is odd" thing applying to the whole family instead of just the children. $\endgroup$
    – JLee
    Jun 8, 2022 at 15:04
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    $\begingroup$ @Eric All twins have the same age but not all people of the same age are twins. $\endgroup$
    – Matthew Wells
    Jun 8, 2022 at 16:08
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I'm going to say the solution is

mother is 21, the kids are 1,2,5,11

This fulfills all requirements. Though,

it means she got her first child at 10 years old. Although controversial, it's certainly possible and not unheard of

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    $\begingroup$ That doesn't satisfy the "sum of ages is odd" requirement. $\endgroup$
    – fljx
    Jun 8, 2022 at 13:52
  • $\begingroup$ @fljx you're right, somehow I misread that the sum of the children's ages needed to be odd $\endgroup$
    – Ivo
    Jun 8, 2022 at 13:53
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This feels like an easy solution or am I missing something?

The mother being 110, one child 21, and four more being 1 year old.
110×21×1×1×1×1 = 2,310
110+21+1+1+1+1 = 135
21+1+1+1+1 > 18
So the ages are 110,21,1,1,1,1

This doesn't exactly work in real life as pointed out in the comments, although there are multiple similar solutions:

77,30,1,1,1,1
70,33,1,1,1,1
55,42,1,1,1,1

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    $\begingroup$ first child when she was 89, then 4 more when she was 108 or 109? $\endgroup$
    – JLee
    Jun 8, 2022 at 13:10
  • $\begingroup$ also, first child at 12 or 13 is very rare AND quads at 55 is also very rare, making both together ultra rare. Quads at 70 or 77 is unheard of. $\endgroup$
    – JLee
    Jun 8, 2022 at 14:24
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I think one answer is:

1,2,3,5,7,11
Because:
1×2×3×5×7×11=2310
Kids are as follows:
1+2+3+5+7=18
The mother is: 11 (kinda weird - she must have adopted at such a young age)
The ages add up to 29
5 children, by the way

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  • $\begingroup$ The question has been updated to specify that the children are her biological children. It's also unlikely that an 11 year old could legally adopt children, so I don't think would have counted anyway unless the question had the lateral thinking tag. $\endgroup$
    – Rob Watts
    Jun 8, 2022 at 20:39

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