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# Father and Son

## Question

The sum of the ages of a father and his son is 66. The father's age is the son's age reversed. Furthermore, the father's age is a multiple of the son's age. How old could they be?

Father: 60 years old

Son : 6 years old

## 56 answers to this question

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Nope, if he is 66 then his son should be 0 (In order to add-up to 66). Of course 0 is not a multiple of 66

One person. He is a father. He is a son. His age adds up to 66. The father's age (66) is a multiple of the son's age (also 66)

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One person. He is a father. He is a son. His age adds up to 66. The father's age (66) is a multiple of the son's age (also 66)

Ok, let me put it this way:

He is a father ==> age of father=66

He is a son ==> age of son=66

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sum of ages of father and son = 132,

The puzzle was

The sum of the ages of a father and son is 66. The father's age is the son's age reversed. Furthermore, the father's age is a multiple of the son's age. How old could they be?

Hope you got the point now.

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Ok, let me put it this way:

He is a father ==> age of father=66

He is a son ==> age of son=66

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sum of ages of father and son = 132,

Hope you got the point now.

Then using the same person 33 would work.

33 + 33 = 66

33 * 1 = 33

33 reversed = 33

60 is not the reverse of 6, 6 is the reverse of 6. If the puzzle had stated "The son's age is the father's age reversed", then 6 and 60 would have worked. 60 reversed is 06, 06 = 6

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The sum of the ages of a father and son is 66. The father's age is the son's age reversed. Furthermore, the father's age is a multiple of the son's age. How old could they be?

1. (xy)+(yx)=66

2. k*(xy)=66

3. x+y=6

{ 24+42=66

{ .====> the answer is the son is 24, the father is 42

{ (4*6)+(7*6)=66 or 24*11/4=66

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Then using the same person 33 would work.

33 + 33 = 66

33 * 1 = 33

33 reversed = 33

60 is not the reverse of 6, 6 is the reverse of 6. If the puzzle had stated "The son's age is the father's age reversed", then 6 and 60 would have worked. 60 reversed is 06, 06 = 6

The puzzle doesn't state there should be two different people. It just says "the sum of the ages of a father and son is 66." not his son.

So, to me 33 could also be an answer.

I don't agree with your statement "60 is not the reverse of 6". Of course it is. Lets start from your last sentence. 6=06. If they are equal, reversing 6 is the same as reversing 06 -- which is 60.

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The sum of the ages of a father and son is 66. The father's age is the son's age reversed. Furthermore, the father's age is a multiple of the son's age. How old could they be?

1. (xy)+(yx)=66

2. k*(xy)=66

3. x+y=6

{ 24+42=66

{ .====> the answer is the son is 24, the father is 42

{ (4*6)+(7*6)=66 or 24*11/4=66

When you speak about multiples, it involves only natural numbers. Otherwise, we will not have prime numbers.

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60 and 6

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Father is 44, son is 22

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Father is 44, son is 22

Nope. The puzzle says "the father's age is the son's age reversed!" ... failed to hold in your case. Try again

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I don't agree with your statement "60 is not the reverse of 6". Of course it is. Lets start from your last sentence. 6=06. If they are equal, reversing 6 is the same as reversing 06 -- which is 60.

I know it's silly nitpicking, but for the sake of argument, I have to agree with Mike H that the age reversal either works only in one direction, or else not at all. If we agree that the "reverse of x" is the result of writing the digits of x in the opposite order, then wouldn't it make sense that, given any input, there's only one correct way to reverse it? If so, then we can define reversal as a function, f(x) = r, where r is the reverse of x. The expected outcome then, is "of course":

f(60) = 06 ... i.e., the son's age is the reverse of the father's age

f(6) = 6 ... i.e., the father's age is NOT the reverse of the son's age

But you say that if we can accept "06 = 6" we can therefore substitute 06 for 6 as the input to the function, f(06) = 60. So the son's age is now "06"? This logic seems faulty. Removing the leading zero is a reasonable simplification, but adding them at will to the input completely breaks down the idea that reversal is a function. For example:

f(0060) = 0600 = 600 ... so the reverse of the father's age is of course 600. Does that make any sense?

I understand the statement was in there for misdirection, and it worked in my case, because I immediately ruled out any single digit possibilities, but I think that basing the misdirection on logic like this actually makes the problem less satisfying.

Edited by Duh Puck

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If we agree that the "reverse of x" is the result of writing the digits of x in the opposite order, then wouldn't it make sense that, given any input, there's only one correct way to reverse it? If so, then we can define reversal as a function, f(x) = r, where r is the reverse of x. The expected outcome then, is "of course":

f(60) = 06 ... i.e., the son's age is the reverse of the father's age

f(6) = 6 ... i.e., the father's age is NOT the reverse of the son's age

But you say that if we can accept "06 = 6" we can therefore substitute 06 for 6 as the input to the function, f(06) = 60. So the son's age is now "06"? This logic seems faulty. Removing the leading zero is a reasonable simplification, but adding them at will to the input completely breaks down the idea that reversal is a function. For example:

f(0060) = 0600 = 600 ... so the reverse of the father's age is of course 600. Does that make any sense?

I understand the statement was in there for misdirection, and it worked in my case, because I immediately ruled out any single digit possibilities, but I think that basing the misdirection on logic like this actually makes the problem less satisfying.

I think you missed one thing to include in the reverse direction -- number of digits. In our case, the number of digits were two ... b/c the sum of the ages of the father and son was 66, and we don't have negative age.

I believe including leading zero's is important. After reversing a number twice (or probably more times, according to the definition), we should be able to go back to the original number. Now, lets consider your examples:

f(60) = 06 = 6 ... I think we already on this one.

f(06) = 60 .... After reversing '60' two times, we got it back. How about if we ignore the leading zero.

f(60) = 06 = 6.

f(6) = 6 ; f(6) = 6 .... we couldn't get the original number. But for some numbers it still holds

f(23)=32 and f(32)=23

From mathematics point of view, this is not well defined. I might be wrong but that is how I see it. By the way, I am not a mathematician some of the terms I just used them loosely.

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I think you missed one thing to include in the reverse direction -- number of digits. In our case, the number of digits were two ... b/c the sum of the ages of the father and son was 66, and we don't have negative age.

I believe including leading zero's is important. After reversing a number twice (or probably more times, according to the definition), we should be able to go back to the original number. Now, lets consider your examples:

f(60) = 06 = 6 ... I think we already on this one.

f(06) = 60 .... After reversing '60' two times, we got it back. How about if we ignore the leading zero.

f(60) = 06 = 6.

f(6) = 6 ; f(6) = 6 .... we couldn't get the original number. But for some numbers it still holds

f(23)=32 and f(32)=23

From mathematics point of view, this is not well defined. I might be wrong but that is how I see it. By the way, I am not a mathematician some of the terms I just used them loosely.

At a minimum, I have to give you credit for making a compelling argument. If we assume that we should be able to get the original number back, then the simplification of removing the leading 0s from the result is superfluous; hence f(60) = 06, end of story, not f(60) = 06 = 6. Of course, we're debating rules that don't really exist in standard mathematics, since we're concerned with visual representation rather than actual value, and thus a lot of the reasoning that both of us have presented doesn't really have much of a foundation to stand on. I still think it's a bit of a stretch to say you can reverse the son's age to get the father's age, but what if we could presume some undefined rule which says something like: "In the case of reversal, the number of digits for both the input and the output has to be the same, and thus the total number of digits is the maximum of the expected input or output."? In that case I think your reasoning could stand. I still don't really agree, but since the concept of "reversal" just doesn't conform to the usual rules, I guess there's some room for latitude.

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At a minimum, I have to give you credit for making a compelling argument. If we assume that we should be able to get the original number back, then the simplification of removing the leading 0s from the result is superfluous; hence f(60) = 06, end of story, not f(60) = 06 = 6. Of course, we're debating rules that don't really exist in standard mathematics, since we're concerned with visual representation rather than actual value, and thus a lot of the reasoning that both of us have presented doesn't really have much of a foundation to stand on. I still think it's a bit of a stretch to say you can reverse the son's age to get the father's age, but what if we could presume some undefined rule which says something like: "In the case of reversal, the number of digits for both the input and the output has to be the same, and thus the total number of digits is the maximum of the expected input or output."? In that case I think your reasoning could stand. I still don't really agree, but since the concept of "reversal" just doesn't conform to the usual rules, I guess there's some room for latitude.

Well, I just thought the leading zeros will enable us to determine uniquely the reverse of a number. f(06)=60 and f(006)=600. Otherwise, I agree with you that if you know the number of digits in advance (like in our case) the leading zeros can be omitted.

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Brhan, I like your posts and work, have to gripe

0=0

00=0

000= guess

But only the first is REAL (proper - as in place holder or number).

You would not say that the answer is a four digit number (0000)

therefore 06 is improper. Like the idea though.

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The only reversable combo are 6/60, 15/51,24/42, 33/33. Adding in the multiple rule clearly makes 6/60 meet all three conditions. I'm sure some might discuss 33/33 and stepchildren, but I'm sticking with 06/60 as my final answer.

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i got 51 and 15 bt 51 isnt a multiple of 15....

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06 and 60, because it's the only one that makes sense, but I have to argue that 6 REVERSED is what? Not 60 in any kind of normal numerical fashion. Also, when writing, depicting or thinking about age, a zero beginning single digits is never used.

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06 and 60, because it's the only one that makes sense, but I have to argue that 6 REVERSED is what? Not 60 in any kind of normal numerical fashion. Also, when writing, depicting or thinking about age, a zero beginning single digits is never used.

Life starting off in double figures ...

My son said his sister was nothing, that is until she was ONE, could not concieve fractions let alone her first year.

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... but I have to argue that 6 REVERSED is what? Not 60 in any kind of normal numerical fashion. Also, when writing, depicting or thinking about age, a zero beginning single digits is never used.

c'mon guys, this is a puzzle. It should be as complicated as possible not to be spot-on easily ...

I surrender now ...

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c'mon guys, this is a puzzle. It should be as complicated as possible not to be spot-on easily ...

I surrender now ...

Are you admitting you are wrong or did you defend the impossible against an infinite enemy - Matrix style?

Maybe a new puzzle from you will do it - yours are usually very good to great.

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Are you admitting you are wrong or did you defend the impossible against an infinite enemy - Matrix style?

Maybe a new puzzle from you will do it - yours are usually very good to great.

thanx 'lost in space' for the comment. I admit that the puzzle could have been worded in a better way.

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thanx 'lost in space' for the comment. I admit that the puzzle could have been worded in a better way.

Easy enough to fix. If it simply said "The son's age is the reverse of the father's age" rather than "The father's age is the reverse of the son's age" then it would be perfectly fine.

People (including myself) may argue that 60 is not the reverse of 6, but I don't think that anyone would argue that 6 is not the reverse of 60.

Edited by rcammoore

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OP states .... Furthermore, the father's age is a multiple of the son's age.

If multiple does not mean integral multiple, the statement gives no information.

Every pair of ages, otherwise, could be described as multiples.

So if you reject that as precise, let's affirm it for all practical purposes.

I would argue that it would give some information, since it would rule out the possibility that the son's age is 0. This is assuming you allow 0 to be counted as an age (and if we're rounding all ages down to the nearest year, why not?), and that the father's age cannot also be 0. Of course, the sum of 66 and reversal conditions already eliminated a zero-year old son, but pedantism has its advantages.

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"The father's age is the son's age reversed" is inaccurate. 6 reversed could mean simply 6 reversed, which is 6 again, 06 reversed, 006 reversed, ect. This sentence should say, "The son's age is the father's age reversed."

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if there both 33 and there father and son thats a bit strange but my guess is son:24 father:42

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