# How old are they?

1. Jun 30, 2004

### Staff: Mentor

Two postmen meet on their routes and they start talking.

Postman A says: "I know you have 3 sons, how old are they?"

Postman B says: "If you take their ages in years, and multiply them together, the result is your age."

A: "That's not enough info"

B: "The sum of the 3 numbers equals the number of windows in that
building over there."

A: "Hmm... that's still not enough."

B: "My middle son is red-haired."

A: "Ah, now I see!"

How old are the 3 sons?

2. Jun 30, 2004

### The Bob

Evo is there something about redheads that we should know but I do not like a redhead is born 3 years after the previous child or something?

Cheers

3. Jun 30, 2004

### vikasj007

apart from something about redhead, i think u should check the question again and see if their is something else which is missing, i mean any numerical value or anything else which you have missed.

4. Jun 30, 2004

### The Bob

Well we do not know the age of Postman A and we cannot assume the amount of windows because the building could have been a shed or a house or even a skyscrap for what we know.

Must be the redheaded child that gives it all.

5. Jun 30, 2004

### chroot

Staff Emeritus
Evo, I believe the dissenters here are correct. I have encountered this puzzle before, and I believe the postman's age is required for a solution.

- Warren

6. Jun 30, 2004

### Monique

Staff Emeritus
Actually no more information is required, the middle red-haired clue should get you started towards a solution! (this question will be easier for biology people )

Last edited: Jun 30, 2004
7. Jun 30, 2004

### Gokul43201

Staff Emeritus
All I can glean from that clue is that a,b,c are all distinct - no two are of the same age.

The first clue tells me that at least one of a,b,c is non-prime.

Last edited: Jun 30, 2004
8. Jun 30, 2004

### The Bob

Middle? Redheaded? This means you know it, right Monique? More hints please.

9. Jun 30, 2004

### Njorl

I think I have it. If you don't want to see the answer, skip my post.

Break the postman's age down to prime factorization. Construct the children's ages from those prime numbers. Each child's age is a product of 0 or more of those numbers (0 prime factors means the child is age 1). Go through all permutations until you find one that fits the logic. You can actulaly eliminate whole catagories of possibilities with general notation.

You can logically eliminate the postman's age being:
prime
a prime squared
the product of 2 primes
a prime cubed
a prime squared times a prime
the product of 3 different primes
a prime cubed times a prime

So, the prime factorization has at least 4 prime factors, and does not take the form "aaab". (a and b are prime numbers)

I tried "aabb" and could not logically eliminate it, but found no logical solutions for a postman 225 years or younger.(edit-you can eliminate this logically, rather than by exhaustion. Just figured it out.)

Trying "aabc" yields a possible solution.

From the logical conditions, you get an equation:

a+a+bc=a^2+b+c, where a,b,c are prime.

The children are aged a^2, b and c. Until the "middle" child clue is given, the possibility remains that they are aged a,a and bc.

Using a=3, b=5 and c=2 yields a possible answer.

Children are 9, 5 and 2, the postman is 90.

There might be other possible solutions for postman of Methuselah's age, but I discounted them.

Njorl

Last edited: Jun 30, 2004
10. Jun 30, 2004

### The Bob

Not to be mean but what about the redheaded child. That was clearly the give away for the postman. Were does it feature in your puzzle?

11. Jun 30, 2004

### Njorl

The redhead is a red herring. The real clue is that there is a middle child. Also, the existance of the fact that there is a middle child is sufficient to clarify a situation from multiple possibilities to one possibility.

The 90 year old postman knew that the sum of the children's ages was 16, and the product of the ages was 90. This meant:

The children could be 3,3 and 10 or 2,5 and 9.

When he learned that there was a middle child, he eliminated the first possibility.

Njorl

12. Jun 30, 2004

### The Bob

Oh ok. I get it. So it could be lots of different numbers? Quality.

Cheers

13. Jun 30, 2004

### Monique

Staff Emeritus
Sorry Njorl, I don't think that answer is correct.. however clever the reasoning :)

How does your reasoning factor in the coming from multiple answers to a single one?

14. Jun 30, 2004

### The Bob

What does the redhead have to do with it?

15. Jun 30, 2004

### Monique

Staff Emeritus
It was the fact that there was a middle child that made it possible for the postman to know the ages of the children. This clue implies two things and Njorl got both of them.

1. Since there is a middle child: there are not twins.
2. The postman must've been doubting between different ages and the fact that there are no twins solved it for him. It is from this fact that you can deduce a mathematical reasoning that will allow you to solve the puzzle without knowing the age of the other postman (since it WAS required for the postman to know the age of the other postman).

16. Jun 30, 2004

### Gokul43201

Staff Emeritus
I think it is A correct answer. The multiple answers come from 3+3+10=2+5+9=16. The middle redhead resolves this.

Perhaps you mean "there is a smaller solution" ?

The only form I can think of is "aaac". This violates Njorl's last eliminated type...but I think he's wrong to eliminate this - the others are okay. Perhaps he used the redhead criterion before the sum degeneracy criterion ?

Let's see...

17. Jun 30, 2004

### Staff: Mentor

Whoa Monique is brilliant!!!

The red headed middle child is the key to finding the mathematical formula, no additional information is necessary to determine the correct ages.

There are not multiple correct answers (well as long as we assume a normal lifespan for the postman and consider normal retirement age.)

Njorl, nope, sorry, but I'm impressed!

Yes, you are on the right track!

18. Jun 30, 2004

### Monique

Staff Emeritus
Um.. how old would the postman be in these answers? :uhh:
These are not possible considerations..

19. Jun 30, 2004

### Gokul43201

Staff Emeritus
Just eliminated aaac's since :

a=2 => no sum degeneracy, easy to show.
a=3 => b<3 => b=2. No degeneracy in this case either.

Perhaps, aabb ?

EDIT : ELiminated all aabb's < 90

Last edited: Jun 30, 2004
20. Jun 30, 2004

### Gokul43201

Staff Emeritus
Why not ?

3*3*10=90=2*5*9. Postman would still be 90 yrs old .

21. Jun 30, 2004

### Njorl

My answer, 90 year postman and children 2,5 and 9 is correct. If you think it is not, I believe that you have made an error. If I am mistaken, please point out the flaw in my reasoning.

Assuming my answer, I will walk through the implications.

I assume 2,5 and 9
He knows his friend is 90, 2x5x9=90
There are a lot of factor triples that can make 90. 2,3,15-3,3,10-1,1,90 even, so he doesn't know yet
the building has 16 windows.
It leaves 2 possibilities, 3,3,10 and 2,5,9
There is a middle child! No twins or triplets
He has eliminated the 3,3,10 possibility and only has the 2,5,9 possibility.
2,5,9

22. Jun 30, 2004

### Monique

Staff Emeritus
Ok.. a 90 year old postman, I'd like to meet him.. I guess you win :tongue2:

23. Jun 30, 2004

### Gokul43201

Staff Emeritus
Njorl, the key may be in recognizing that you can multiply by 1 and not change the product.

24. Jun 30, 2004

### Monique

Staff Emeritus
Permutations might also lead you to the following answer: 1,5,8 (with 2,2,10 being the twin), but it is all about the logical behind the question

25. Jun 30, 2004

### Staff: Mentor

Monique is correct.

Njorl, if we stretch it and say that the postal service has 90 year old postmen, then, yes, you have a correct answer. :tongue2:

Remember, part of the information you've been given to work with is that it is a working postman, so under normal circumstances we would assume the answer would be less than 65.

Here's the explanation.

The kids are 1, 5 & 8 because they cannot be 2, 2 & 10.

The product of both are 40 and the sums are 14
The basis for that solution is as follows
The postman’s first statement tells us that x*y*z=A; the
Postman’s second statement tells us that x+y+z=B; and the
Third statement tells us that x does not equal y does not equal z
We know that the three ages are different because the “middle”
Son has red hair, which implies that none of the children are twins
Important note: It is not the fact there are no twins, but the fact
That the postman could not figure out the answer UNTIL he knew
That there were no twins that was essential, thus
There must be only two different sets of age choices for the kids.
(one with twins; one without twins) that produce equivalent A’s and
B’s. With a little bit of work, you can discover that only one
Solution is possible: the kids must be 1, 5 & 8 because a
Solution with the same A & B involving twins (2,2,& 10)
Is ruled out by the middle son with red hair.