# How old are my 3 children?

1. Dec 23, 2004

### T@P

ignoring the fact that i have no children, here is the question:

(this is really a dialogue between dimitry and fred)

dimitry: hi fred

fred: hi

dimitry: hows it going?

fred: fine, you?
...
(after 1 hour of small talk)

dimitry: hey fred, how old are T@P's children?

fred: if you multiply their ages (in years), you get the age of Tom.

dimitry: hmmmm thats not enough. any more clues?

fred: if you add their ages (in years) you get the number
of windows in that building over there.

dimitry: cmon fred thats still not enough. how old are they?

fred: T@P's middle son has blue hair (like some anime character!)

dimitry: ok thanks for telling me theire ages.

fred: bye

dimitry: bye

so how old are they?

some small clarification: dimitry does not know toms age. also i have 3 children.

2. Dec 23, 2004

### Rogerio

Select to see...
the ages are 0, 1, & 2

3. Dec 23, 2004

### T@P

i like to think of my children as alive you know.
also, tom cannot be 0 years old, and, well, just no they cant be 0
:)

4. Dec 23, 2004

### Rogerio

If tom was born last month, then he would be ___ years old ...:-)

Tell me you !

5. Dec 23, 2004

### T@P

true.... but tom wouldnt, because he is a college drop out that sells ilicit materials to little children... so he isnt 0 years old :)

6. Dec 23, 2004

### Lyuokdea

The children are 8,5,1 years of age, tom is 40 and there are 14 windows in the building.

My process for finding the solution was mainly brute force though and depended on giving tom a resonable age, because I assume there are probably infinately many solutions for any two numbers of this type. If somebody could show a mathematical proof of this which finds all the right numbers, I would find it very interesting. The best way to find a lot that I know of would be to program it, which I might try later, I'll post my code if I get around to writing it.

~Lyuokdea

7. Dec 24, 2004

### mahesh_2961

hmm.. is it 1,3 and 6 .

mahesh

8. Dec 24, 2004

### ceptimus

If Dimitri really doesn't know Tom's age (i.e Tom can be any age from 1 up to the oldest a human can live, 125? then all he knows is that the sum of the ages is equal to the number of windows. The final clue tells him that no two of the children have the same age.

Are you sure Dimitri doesn't know Tom's age? I think the puzzle has multiple solutions unless he does.

1, 2, 3 = 6 windows (could have been 1, 1, 4 or 2, 2, 2 etc. before the final clue)

1, 2, 4 = 7 windows (could have been 2, 2, 3 or 1, 3, 3 etc. before the final clue)

I'm ignoring ages of 0 (i.e. less than one year old, but if you include them, that makes the puzzle seem even harder to me)

9. Dec 24, 2004

### T@P

semi-major hints below:

the point of the solution is that dimitry *does not have enough information* until he finds out that the children all have diferent ages. that is essentially the point of the puzzle, because otherwise almost any solution may work

so ceptimus, the problem comes down to finding two sets of (3) numbers, 1 set where two are distinct and the other where all three are, and theyre sum is the same, as is the number when you multiply them.

10. Dec 24, 2004

### ceptimus

I don't understand why the product of the three numbers has to be the same for both sets of numbers, unless Dimitri knows Tom's age.

11. Dec 24, 2004

### T@P

i dont exactly feel as though it would be fair if i told this to everyone, so it is in white:

dimitry, until he heard that the middle son is unique, had several solutions to this puzzle in his head. now since he is incredibly smart, what made him find out the solution was that there are 3 distinct ages of the children. so backtracking a little bit,
x*y*z = tom and x + y + z = window according to what dimitry thinks. now once he knows that x != y, then there must be a different set of numbers, a * a * b = tom and a + a + b = windows, and by finding out that the numbers are all different, he chooses the x and y one and knows the answer. becasue that is true, there must then exist these numbers, such that theire multiplication is the same and theyre sum is, and yet they are different integers. this basically tell you how to do the problem, and yet it doesnt tell you the answer, since finding them is almost as hard as knowing how to do the problem.

i hope i answered your question, and im sorry everyone else i dont have time to check your answers, but by backtracking using the way i metioned above you should be able to find out yourself.

12. Dec 24, 2004

### Gokul43201

Staff Emeritus
There are a few reasonable solutions :

2,5,9 : sum=16, product = 90, Tom's pretty old (not 3,3,10, due to middle child)

1,5,8 : sum = 14, product = 40 (not 2,2,10)

I think these are the only solutions - besides Rogerio's family of solutions involving 0 ages - where the product is small enough to be a normal human's age.

PS : I'm assuming dmitri knows Tom's age...else this problem wouldn't be sensible.

Last edited: Dec 24, 2004
13. Dec 24, 2004

### Rogerio

Well, T@P has said dimitry does not know Tom's age , and I think there is only one solution...

Of course, 0 years old is a valid age. So, initially we just know the 3 ages (age1, age2, age3) are non-negative integers.

The first clue is : age1*age2*age3 = tom's age (dimitry doesnt know the toms age)

As fred accepted it was not enough, the only conclusion is
"Tom is not 1 year old and the 3 childreen are not 1 year old" ,
and nothing more.

Note that, at this point, they could be age1=1, age2=1, and age3=Tom's age.

The second clue (age1+age2+age3 = #windows of the building) was not enough, so there is more than a way to partition that number into 3 numbers.

The third clue (there is a middle age) was enough, so, there is only one way to partition that number (the sum of ages) into 3 different numbers.

The unique number which can be partitioned into 3 different numbers, in only one way, is 3.

So, the ages are 0,1,2.

14. Dec 24, 2004

### T@P

look tom cant sell ilict materials if hes 0 :) besides there are better solutions, so dont get too excited about the 0 one. also i think gokus are right.

I cant figure out why there are two. it could be because then hes too old, but im not sure. I was wondering how you came up with those numbers, because i spent like 1 hour trying to thinko f a way to find them other than guessing, and im not getting very far.

15. Jan 7, 2005

### T@P

i dont want to sound pushy, but i am really quite interested in how you (goku43201) managed to find the numbers (answers)? if someone can tell me a feasable method, or if goku43201 can tell me, that would be excellant

16. Jan 7, 2005

### Gokul43201

Staff Emeritus
Last edited: Jan 7, 2005
17. Jan 7, 2005

### Alkatran

Data:
t > 0
a*b*c = t
a+b+c = x
x is known, t is unknown
a < b < c

Calculated and Assumed Data:
a*b*c > 0
a > 0, b > a, c > b
b > 1, c > 2
t > 1*2*3-1 = 5
t > 5

Logic:
Now, them sum of these ages, when giving X, must have one or more solutions where the ages are equal and one where they aren't. We can stop looking once we hit this number by assuming there is only one answer that we need. We will start at the lowest possible number, given the minimums on the ages.

x=6:
2,2,2; 3,2,1; 3,3,0
6 will work

x=7: 2,2,3; 1,2,4; 1,3,3; 1,1,5;
7 will work

x=8: 1,2,5; 2,3,4; 1,3,4
8 won't work

x=9: 1,2,6; 1,3,5

Any answer above x=7 is not feasible because it can be of the form 1,2,(x-3) and 1,3,(x-4)
The limit is at 7 because 7-4 = 3, and for 6: 6-3 = 3, eleiminating one of the answers via a!=b!=c

All in all:
Some arrangement of a 1 year old, a 2 year old and a 3 or 4 year old

Last edited: Jan 7, 2005
18. Jan 9, 2005

### Rogerio

Why?
If you were born 6 months ago, then you would be 0 years old.

19. Jan 9, 2005

### Alkatran

Because Tom's age is more than 0 (he can sell drugs) so none of the ages can be 0. Thus a*b*c > 0

20. Jan 9, 2005

### Rogerio

However, according to you, Tom would be 6 ou 8 years old...