[Guest]

A mother has 6 children.

The sum of their ages is the same as her age.

The product of their ages is 77,760.

They are all different ages.

The sum of the difference between their ages is the same as the mothers age backwards.

What are their ages?

[Guest]

What do you mean by "The sum of the difference between their ages"?

If the ages were lets say: 1 2 5 10, then the differences are 1 3 5 (the difference between each child and the child after him) or do you mean the sum of the difference between every possible combination of two children?

Edited August 18, 2011 by Anza Power

[Guest]

From the way it's worded, "The sum of the difference between their ages" suggests it's a consistent interval, or it would've been stated as 'differences?' Not entirely sure about this one, but it seems logical.

So, is it:

The sum of: X1 - X2 - X3 - X4 - X5 - X6 = D?

OR

The sum of:

X1 - d = X2

X2 - d = X3

X3 - d = X4

X4 - d = X5

X5 - d = X6

[Guest]

It turns out that the "sum of the difference between their ages" is actually

"the sum of the difference between each child and the child after him"

and not "the sum of the difference between every possible combination of two children".

The latter would lead to a non-existent solution.

So you only need to subtract the oldest minus the youngest to get that difference.

So, my solution is (1, 4, 5, 9, 16, 27) which is unique.

To find it I used the following Python code:

[ (a,b,c,d,e,f) for a in range(70) for b in range (a+1,70) for c in range(b+1,70)

for d in range(c+1,70) for e in range(d+1,70) for f in range(e+1,70) if a*b*c*d*e*f==77760

and f-a == int(''.join(reversed(str(a+b+c+d+e+f)))) ]

I assume that all childred are < 49 years old, and the mother is < 100.

Edited August 18, 2011 by ursht

[Guest]

It turns out that the "sum of the difference between their ages" is actually

"the sum of the difference between each child and the child after him"

and not "the sum of the difference between every possible combination of two children".

The latter would lead to a non-existent solution.

So you only need to subtract the oldest minus the youngest to get that difference.

So, my solution is (1, 4, 5, 9, 16, 27) which is unique.

To find it I used the following Python code:

[ (a,b,c,d,e,f) for a in range(70) for b in range (a+1,70) for c in range(b+1,70)

for d in range(c+1,70) for e in range(d+1,70) for f in range(e+1,70) if a*b*c*d*e*f==77760

and f-a == int(''.join(reversed(str(a+b+c+d+e+f)))) ]

I assume that all childred are < 49 years old, and the mother is < 100.

oh!!

excellent answer

but can you say which language you used for programming

[Guest]

Here is a more mathy approach, though ending with trial-and-error, I'm afraid:

Prime factorization of 77760 = (2^6) * (3^5) * 5

Thus we need to divide (2^6) and (3^5) into a total of 5 unique numbers. Each must contribute at least two numbers and at most three, since both 64 and 243 are too old for a child. Hence:

(2^6) = 2 * 4 * 8 OR 1 * 4 * 16 OR 4* 16

(3^5) = 9 * 27 OR 1 * 9 * 27 (81 is also too old for a child)

Which yields only 3 possibilities:

2 4 5 8 9 27

1 4 5 9 16 27

1 2 5 9 27 32

Only the second works, adding up to 62 while the sum of differences adds to 26.

This means that the woman had a child when she was 61, which, sure, we'll allow. :-)

Edited August 18, 2011 by ep4169

[Guest]

You guys did some really good calculations.

<Ursht> is right with the way to calculate the sum of the difference between their ages, I love the easy way " oldest minus youngest"

Also I like the math approach from <ep4169>

But nobody has the right ages of the children yet, which I have to admit is my fault !

I was a bit overtired when I submitted this puzzle and made a mistake, the sum of the difference of their ages is actually not the mothers age backward - it's 14.

I was 1 off, so I give you guys this extra clue which should make it easy to solve the riddle

[Guest]

You guys did some really good calculations. <Ursht> is right with the way to calculate the sum of the difference between their ages, I love the easy way " oldest minus youngest" Also I like the math approach from <ep4169> But nobody has the right ages of the children yet, which I have to admit is my fault ! I was a bit overtired when I submitted this puzzle and made a mistake, the sum of the difference of their ages is actually not the mothers age backward - it's 14. I was 1 off, so I give you guys this extra clue which should make it easy to solve the riddle

There are a few more options that those given yet. Remember that any combination of six 2's, five 3's, one 5, and any number of 1's you need is valid.

Child        	A    	B    	C    	D    	E    	F

Factor 1  	4    	5    	9  	3    	3    	2

Factor 2  	4    	3    	1	2    	1    	1

		-----------------------------------------------

Age 		16  	15    	9   	6  	3    	2


Product        	= 77,760

Age of Mom  	= 51

Sum difference  = 14

[Guest]

Running the program again, considering the new corrections,

the result is either (1, 6, 8, 9, 12, 15) or (2, 3, 6, 9, 15, 16).

Both are valid. No other clue is given. So we don't have a unique answer.

Here is the Python code:

[ (a,b,c,d,e,f) for a in range(70) for b in range (a+1,70) for c in range(b+1,70)

for d in range(c+1,70) for e in range(d+1,70) for f in range(e+1,70)

if a*b*c*d*e*f==77760 and f-a == 14 ]

Here, I assume that all children are < 70 years old.

Edited August 18, 2011 by ursht

[Guest]

Here is one more clue and then you'll know the right answer for sure. If you add the ages of two kids at a time starting with the youngest to the oldest the sums all end with seven.

[Guest]

Running the program again, considering the new corrections,

the result is either (1, 6, 8, 9, 12, 15) or (2, 3, 6, 9, 15, 16).

Both are valid. No other clue is given. So we don't have a unique answer.

Here is the Python code:

[ (a,b,c,d,e,f) for a in range(70) for b in range (a+1,70) for c in range(b+1,70)

for d in range(c+1,70) for e in range(d+1,70) for f in range(e+1,70)

if a*b*c*d*e*f==77760 and f-a == 14 ]

Here, I assume that all children are < 70 years old.

One of your answers is right!

[Guest]

The problem has been solved, the winner is <ursht>

[spoiler=The ages of the kids were 1,6,8,9,12 &15; The mom's age is 51 , the

difference in ages is 14 and the product = 77760

Edited August 21, 2011 by BusyMom