From Newsgroup: rec.puzzles

From `536 Puzzles & Curious Problems` by Henry Ernest Dudeney.

A man had nine children, all born at regular intervals, and the sum of the squares of their ages was equal to the square of his own. What was the age
of each? Every age was an exact number of years.
--
David Entwistle
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From Newsgroup: rec.puzzles

In article <10l2aao$na1l$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

A man had nine children, all born at regular intervals, and the sum of the >squares of their ages was equal to the square of his own. What was the age >of each? Every age was an exact number of years.

Did they all have the same mother?

-- Richard
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From Newsgroup: rec.puzzles

On 24/01/2026 11:27, David Entwistle wrote:

From `536 Puzzles & Curious Problems` by Henry Ernest Dudeney.

A man had nine children, all born at regular intervals, and the sum of the squares of their ages was equal to the square of his own. What was the age
of each? Every age was an exact number of years.

If the minimum age is 'a' and the gap is 'd', then the father's age is
'f', then ...

f = sqrt(a^2 + (a+d)^2 ... (a+8d)^2, where f is an integer.

I'm sure Dudeney has an elegant solution, but that is beyond me.

By brute force ...

$ ./a.out
Childrens' ages: 2, 5, 8, 11, 14, 17, 20, 23, 26. Gap = 3. Father's age: 48 Childrens' ages: 4, 10, 16, 22, 28, 34, 40, 46, 52. Gap = 6. Father's
age: 96
Childrens' ages: 6, 15, 24, 33, 42, 51, 60, 69, 78. Gap = 9. Father's
age: 144

Probably the age gap being a multiple of 3 is significant.

The father is obviously not 144 years old.

If the father was 96, then he had his youngest child at 92 - which is
rather good going, but unlikely.

So, the answer has to be every 3 years: 2, 5, 8, 11, 14, 17, 20, 23, 26
and the father is 48.


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From Newsgroup: rec.puzzles

On 24/01/2026 12:20, Richard Tobin wrote:

In article <10l2aao$na1l$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

A man had nine children, all born at regular intervals, and the sum of the >> squares of their ages was equal to the square of his own. What was the age >> of each? Every age was an exact number of years.

Did they all have the same mother?

I'd hope so. Can you imagine how expensive that would be otherwise?



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From Newsgroup: rec.puzzles

In article <10l2ds9$pegn$1@dont-email.me>,
Richard Harnden <nospam.harnden@invalid.com> wrote:

If the father was 96, then he had his youngest child at 92 - which is
rather good going, but unlikely.

That's why I asked about the mother - the first and last children
were born 48 years apart. Is that (ahem) conceivable?

-- Richard
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From Newsgroup: rec.puzzles

In article <10l2ds9$pegn$1@dont-email.me>,
Richard Harnden <nospam.harnden@invalid.com> wrote:

f = sqrt(a^2 + (a+d)^2 ... (a+8d)^2, where f is an integer.

A way to tidy this up is to let a be the _middle_ term and look at

(a-4d)^2 + ... + a^2 + ... + (a+4d)^2 = n^2

Then all the middle terms of the squares cancel, and we get let with just

9a^2 + 60d^2 = n^2

The left-hand side is a multiple of 3, and so if it's a square then it's
a multiple of 9, and so d itself is a multiple of 3. So let's set d =
3e and n = 3m, and we get

a^2 + 60e^2 = m^2

We can see fairly quickly that 196 and 256 are squares differing by 60,
so let a = 14, e = 1, m = 16, and we get your n = 48 answer.

Gareth
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