What Ages Solve the Postmen's Riddle?

[Evo]

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?

 

[The Bob]

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?

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

The Bob (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.

 

[The Bob]

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.

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.

The Bob (2004 ©)

 

[chroot]

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

 

[Monique]

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 )

 

[Gokul43201]

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.

 

[The Bob]

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 )

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

The Bob (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

 

[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?

The Bob (2004 ©)

 

[Njorl]

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?

The Bob (2004 ©)

The redhead is a red herring. The real clue is that there is a middle child. Also, the existence 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

 

[The Bob]

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

Cheers

The Bob (2004 ©)

 

[Monique]

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?

 

[The Bob]

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?

What does the redhead have to do with it?

The Bob (2004 ©)

 

[Monique]

What does the redhead have to do with it?

The Bob (2004 ©)

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).

 

[Gokul43201]

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?

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...

 

[Evo]

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).

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!

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.

Yes, you are on the right track!

 

[Monique]

I think it is A correct answer. The multiple answers come from 3+3+10=2+5+9=16.

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

 

[Gokul43201]

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

 

[Gokul43201]

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

Why not ?

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

 

[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.

Two postmen meet on their routes and they start talking.

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

I assume 2,5 and 9

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

He knows his friend is 90, 2x5x9=90

A: "That's not enough info"

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

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

the building has 16 windows.

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

It leaves 2 possibilities, 3,3,10 and 2,5,9

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

There is a middle child! No twins or triplets

A: "Ah, now I see!"

He has eliminated the 3,3,10 possibility and only has the 2,5,9 possibility.

How old are the 3 sons?

2,5,9

 

[Monique]

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

 

[Gokul43201]

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

 

[Monique]

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

 

[Evo]

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

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.

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.

 

[Gokul43201]

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

Knew there would be a sneaky 1 around somewhere. Had just eliminated aabc with 1's. aaabc would have been next (40=2*2*2*5*1)...but Monique let the cat out !

I believe Njorl (as did I, initially) made the mistake of not recognizing the possibility of a 1 year old. Since 1 is neither prime nor composite, (it is trivial in the factorization) it slipped past the mathemtical net that Njorl made to catch the ages.

 

[Gokul43201]

Monique, how did you solve it ?

 

[Njorl]

Ah.

As a federal employee, I know there is no federally mandated retirement age. I was limiting myself to 128 (2^7) as the limit of human life.

I just noticed that Gokul was correct that I erroneously eliminated a prime factorization of the form aaab for the age of the postman from the possibilities. I had been considering 1 year olds, I guess I was just sloppy. It turned out to be the best answer, a postman who was 2x2x2x5 years old. (a=2, b=5)

I suppose a 40 year old mailman is a bit more believable than a 90 year old mailman.

Njorl

 

[Evo]

Well, here is confirmation that Njorl did indeed get the only other solution which would still be in a human's lifetime, although it is a very eldery postman.

More detailed explanation for the Postman Riddle:
The most important feature to recognize in solving this puzzle is that Postman A needed to know that Postman B did not have twins in order to solve the puzzle. That means that there must be two possible solutions that satisfied the "age in years" and "number of windows" conditions of the problem. He needed to rule out twins to solve the problem. To summarize, there must have been two potential solutions with 3 son's ages having:

(i) equal products (the age condition)

(ii) equal sums (the windows condition)

(iii) one involving twins and one not.

If you really wanted to be fussy, since a house cannot have a fraction of a window, it also means that the ages of the sons must all be in integers (whole numbers) and not involve fractional ages.

The one solution to the problem that satisfies all of these conditions is (1, 5, 8) which has the same sum (14) and the same product (40) as another possible solution involving twins (2, 2, 10). There is just no other solution that works involving a postman's age less than 90 years old. That is, another solution that works mathematically is 2-5-9 and 3-3-10, but that would make the postman 90 years old.

 

[Njorl]

I got to get me a look at article 2 of Volume IV of "Georgia Backroads" magazine.

http://www.georgiahistory.ws/county_articles/vol4.asp

I'm sure he's at least 90. And you know, Georgia's just full of 16 window buildings! I think that's what is meant by Georgian architecture - exactly 16 windows ... yeah, that's the ticket!

Njorl

 

[NateTG]

Be happy it wasn't "My youngest son has red hair"

Then the choices would be:
36:
2,2,9
1,6,6

40:
2,2,10
1,5,8

or

90:
3,3,10
2,5,9

 

[Brennen]

why couldn't you have 2 children of the same age who weren't twins? they could b 3, 3, 10 with one of the 3 year olds being older than the other. even if they are twins, one of them would still be older, and therefore the middle child.

 

[Gokul43201]

Unless the postman worked in Utah, this would be highly unlikely. I'll let the biology folks tell us the least time between successive births (I know there are several factors involved, including how long you breast-feed)...but I'm sure that 2 births from one woman, less than a year apart is virtually unheard of.

Or he could always divorce a pregnant wife; re-marry and make his new wife pregnant in short order...

Or there could be illegitimate children...

Clearly, we're nitpicking and getting beyond the intended realm of the puzzle.

 

[TENYEARS]

You really tripped them up with this one.

 

[PureEnergy]

I don't know how rare it is, but I'm 10 months older than my younger brother.