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Too Many Children

Math brain teasers require computations to solve.


Fun:*** (2.25)
Difficulty:*** (2.8)
Submitted By:stormtrooper


A man has nine children born at regular intervals. The sum of the square of their ages is equal to the square of his own age. What are the ages of his children?

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Posted by jimeich1 on May 16, 2005

Could you show the math on that?

Posted by darthforman on May 16, 2005

Posted by annie9698 on May 16, 2005

no clue how you came up with that

Posted by beadbabe on May 16, 2005

I didn't even know where to begin! Good one!

Posted by bigSWAFF_69_ on May 16, 2005

good job, all u havta do is find the square of their ages, add 'em together, then u have his age. After that u just havta figure out that the interval is 3. I did not figure this out, but after reading the answer, i easily figured out how to do it. Good job!

Posted by brianz on May 16, 2005

Can you prove that that's the only solution?

Posted by cnmne on May 16, 2005

Why the low popularity rating? I guess some people do not like math.

Posted by midnight_blue on May 17, 2005

explain explain! dun leave us blank n curious on how to arrive at that solution...

Posted by tick on May 17, 2005

Very well put, I liked this teaser.

Posted by smurfy on May 17, 2005

Please explain. I understand the answer, but how do you come up with the answer? I had too many variables when i tried to set up this problem. (A+B+C+D+E+F+G+H+I)^2 = J^2 Also, the first step is to take the square root of both sides, so why do they need to be squared? I think it would be easier to find if you said that the children had three years in between them and you gave us the age of the children.

Posted by jennysugars on May 19, 2005

I liked the challenge to my brain even though I never figured out the answer. After I looked at the answer, I still never figured out how you arrived at it. I would love to see a formula for how you set this one up! It's a devil of a teaser!

Posted by on May 19, 2005

I agree with brianz. There is more than one solution. I still had fun figuring that out. Here's the formula:


(continues to the ninth child increasing the interval factor by 1)
All that simplifies to:


Posted by silly07 on May 20, 2005

PPPPPlease tell us how u got it

Posted by libra0890 on May 22, 2005

nooooooooo 48^2 is 2304, and 26^2 is not 2304 therefore it does not work.

Posted by on May 25, 2005

There is no closed form answer to this question.

The answer is a classic hunt and peck searching for the square root of a sum that happens to be a round number. This is because there is only one equation involving 3 variables, the man's age at the birth of first child, the interval between births. and the interval between the last birth and the current age.

Posted by jimeich1 on May 26, 2005

I got the same formula as you did.
What puzzles me is that the normal form for the square of a binomial is:

In this case, the formula is not in
that form. That is, in the polynomial,
9a^2+72ab+204b^2, the sum of the
square roots of the outer terms
doubled is NOT equal to the middle

Can anyone explain?


Posted by ane92590 on May 28, 2005

this question is not worded correctly. in order for that answer to be true, the question must state that all the numbers are whole numbers. otherwise, there is an infinate amount of solutions.

Posted by elvenberserker on May 31, 2005

I'm sorry to tell all you guys, but that was way to easy to be called a hard teaser. I used paper and pencil and got the answer in about 3 minutes only. maybe you should just have paid attention to your math teachers.

Posted by jimeich1 on Jun 03, 2005

Hey elvenber,

Since I am one who must have slept through math class, could you answer my question posted above?

We would all appreciate it.

Posted by Darveed on Jun 06, 2005

Good teaser. totally stumped.

Libra...the idea is 48^2 = 26^2+23^2+20^2 etc etc. Thats why it says the sum of

Posted by anakinschick on Jun 06, 2005

what the heck. thats all im gonna say

Posted by solidtanker on Jun 07, 2005

The equation that many of you got stuck with is called a Diophantine equation (after Diophantus the 3rd century mathematician). A diophantine equation is an equation with more than one variable (obviously) whose solutions must be integers. These equations cannot be solved so easily. You have to eliminate certain cases, and through trial and error and alot of deduction, you come upon an answer which satisfies the said requirments. For those who are about ready to start pulling their hair out, don't despair. After all, Fermat's last theorem was a diophantine equation, and it wasn't solved (or rather proven unsolvable) for about 350 years!!!

Posted by on Jun 07, 2005

There is a second solution:

Children are 4, 10, 16, 22, 28, 34, 40, 46 and 52.

The man's age is 96.

Not as likely a scenario as the posted answer, but certainly within the realm of possibility.

As far as I can tell there are no other solutions with whole numbers and the man's age under 100.

Posted by on Jun 07, 2005

There are also oddball solutions, such as:

3 sets of triplets, ages 11, 11, 11, 23, 23, 23, 35, 35, 35. Man is age 75.

Or, the degnerative:

a set of nontuplets (???), all age x, with the man age 3x. (e.g. 15 and 45)

Or, triplets aged 7,7,7, 25, 25, 25, 43, 43, 43 and the man age 87.

You could argue that 3 sets of triplets born at equal intervals meets the criteria.

Posted by solidtanker on Jun 07, 2005

Those are possiblilities only if you disregard the statment that the children were born at regular intervals. But there is only one solution which makes sense for this problem as it is stated, with the ages being integers.

Posted by solidtanker on Jun 07, 2005

Actually I stand corrected. Your first alternate solution is just as viable. By multiplying both sides by a factor of 2, the solution still makes sense. But it would not work for a factor of three and more (in the real world). Good job!

Posted by sk8ergrl on Jun 13, 2005

how are you supposed to get this when you don't know how old any of the kids are??????/ I don't get it at all

Posted by paul726 on Dec 08, 2005

I too came up with Robo06's equation. It really comes down to trial and error, which is fine, but I definitely prefer having a way to use some math to at least reduce the possibilities to a reasonable volume.

Posted by pating on Jun 19, 2007

another one of those teasers that dont actually have a real solution. more like a trivia question.
i like my math teasers to have a solution using math or a little logic, not relying solely on trial and error.

Posted by javaguru on Feb 09, 2009

Yeah, while my heart is with Pating's comment, it was pretty cool. I wouldn't have seen it in trick, so I'm glad it was here.

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