*by Michael S. Kaplan, published on 2006/07/06 14:42 -04:00, original URI: http://blogs.msdn.com/b/michkap/archive/2006/07/06/658161.aspx*

Seems like everyone wants a piece of Jason's interview question:

*Two MIT math graduates bump into each other at Fairway on the upper west side. They hadn't seen each other in over 20 years.The first grad says to the second: "How have you been?"Second: "Great! I got married and I have three daughters now"First: "Really? how old are they?"Second: "Well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there.."First: "Right, ok.. oh wait.. hmmmm.., I still don't know"second: "Oh sorry, the oldest one just started to play the piano"First: "Wonderful! my oldest is the same age!"*

*Problem: How old are the daughters?*

First of all, I find myself very sympathetic to Jason's answer to this question. :-)

Personally, I think this is a good question to help determine if someone is having trouble with their work/life balance.....

Anyway, here is my logic.

The answer I have seen in the past is the 8, 3, 3 one that Adam mentions on Gretchen's blog, as well as 2, 4, 9. Since twins are never mentioned, that second answer also works just as well.

But I think these are both wrong -- or at least if not wrong, at least not proven.

The only clues not "used" yet:

"Fairway on the upper west side"

"same as the number on that building over there"

I think most people assume it is a golf course fairway, but that is unlikely (there are no golf courses on the upper west side).

Assuming the capitalization is intentional, they probably mean Fairway Market (which is on the upper west side), which is on 2127 Broadway.

(they could also be thinking about the street number, which would be either 74th or 75th, but let's assume the actual street number for now).

So we are actually looking for three ages that add up to twelve -- so both of those answers fail now.

Maybe there is some other number on Fairway Market?

Or maybe they mean the other Fairway Market, the one on 2328 12th Avenue. But since 2 + 3 + 3 + 8 == 16, that can't be it either. Let's stay on the Upper West Side.

Ok, there is a sign for the nearby subway stop -- 1, 2, 3 at 72nd St. And 1 + 2 + 3 + 7 + 2 == 15, which is the same as 2 + 4 + 9.

So I will guess that the ages are 2, 4, and 9.

How'd I do? :-)

# **Michael S. Kaplan** on 6 Jul 2006 2:50 PM:

A friend of mine who used to live in New York just mentioned that there is a "1950" cornerstone on the building, which might also be the sign that was meant.

That would also add up to 15. Independent confirmation? :-)

That would also add up to 15. Independent confirmation? :-)

# **Joe** on 6 Jul 2006 4:06 PM:

The key to the question is the 'hmm... I still don't know.'

To find the number of the 'building over there' you need to find what two(or more) sets of numbers have the same product (72) and sum.

The only qualifying ages are 6,6,2 and 8,3,3. (All of the other sets have a unique sum).

The answer is 8,3,3 because it's unlikely that someone would refer to an older twin as 'my oldest.'

To find the number of the 'building over there' you need to find what two(or more) sets of numbers have the same product (72) and sum.

The only qualifying ages are 6,6,2 and 8,3,3. (All of the other sets have a unique sum).

The answer is 8,3,3 because it's unlikely that someone would refer to an older twin as 'my oldest.'

# **Michael S. Kaplan** on 6 Jul 2006 4:13 PM:

I understand that, but I have a hard time buying an arbitrary sign that we do not know the source of that happens to have a number on it.

Anyone from NYC able to suggest a sign at Fairway Market that has numbers that add up to 14?

Anyone from NYC able to suggest a sign at Fairway Market that has numbers that add up to 14?

# **Anthony Mills** on 6 Jul 2006 4:26 PM:

On the sign is the number 14, not numbers that add to 14. I'm sure somewhere at Fairway Market there's a sign with "14" on it. The answer 2, 4, 9 does /not/ work just as well because it adds up to the unambiguous sum 15. If it was 2, 4, 9, the second MIT grad would have not hesitated at all.

# **Michael S. Kaplan** on 6 Jul 2006 4:41 PM:

Perhaps, it just doesn't feel as satisfying to me to not have that piece of it locked down a bit better. :-)

# **Sebastian Redl** on 6 Jul 2006 7:18 PM:

> How'd I do? :-)

You failed to explain how knowledge of what numbers might be on a house near a place called Fairway has anything to do with logic.

I dare to say that there's more than one Fairway with an upper west side in the world, or even the US. Your assumption that it's about Fairway Market in New York is quite simply invalid.

You also, rather ironically, made a mistake in the semantic interpretation of the riddle. The formulation does not permit the digits of the number on the house to be added up.

That said, it's a creative way of thinking about the problem.

You failed to explain how knowledge of what numbers might be on a house near a place called Fairway has anything to do with logic.

I dare to say that there's more than one Fairway with an upper west side in the world, or even the US. Your assumption that it's about Fairway Market in New York is quite simply invalid.

You also, rather ironically, made a mistake in the semantic interpretation of the riddle. The formulation does not permit the digits of the number on the house to be added up.

That said, it's a creative way of thinking about the problem.

# **Michael S. Kaplan** on 6 Jul 2006 7:23 PM:

Hi Sebastian,

I think I succeeded in a slightly more well rounded approach to the problem that goes beyond games with numbers!

I think I succeeded in a slightly more well rounded approach to the problem that goes beyond games with numbers!

# **Vlado Klimovský** on 7 Jul 2006 12:02 AM:

Michael, why don't you just admit that you did not "get" it? Three commenters independently explained why your interpretation and solution is simply wrong, and you still hang on to it. I can only assume that you are doing it intentionally :-)

BTW, I've seen this riddle in another language, and of course without any mention of Fairway or upper west side whatsoever...

BTW, I've seen this riddle in another language, and of course without any mention of Fairway or upper west side whatsoever...

# **Bruce Rusk** on 7 Jul 2006 3:02 AM:

In response to Joe's claim that "The answer is 8,3,3 because it's unlikely that someone would refer to an older twin as 'my oldest.'":

Two siblings of the same age need not be twins; they can be born, say, ten months apart. If it were in the window after the birthday of the younger of the two siblings but before the birthday of the next, they would be the same age (in whole years).

Two siblings of the same age need not be twins; they can be born, say, ten months apart. If it were in the window after the birthday of the younger of the two siblings but before the birthday of the next, they would be the same age (in whole years).

# **Ian Griffiths** on 7 Jul 2006 5:02 AM:

I'll take a crack at explaining it one more time. Maybe verbosity will out. :)

The point about the house number is simply this:

We know the guy trying to work out the ages can see the number. We therefore know that *he* knows exactly what number it is. We don't know what the number is, but that doesn't matter. The important information revealed here is that the guy knows what the number is.

Why is that important information? Well, we know that even though he knows this number, that's insufficient information for him to work out the ages.

So what information do we have at this stage? We know the product of the ages is 72. And we know that even if you know the sum of the ages as well as the product, that's not enough to work out the individual ages.

I believe these are all the possible ages that give a product of 72 where the numbers are all under 20, and I've shown the sum of the ages next to them:

1, 8, 9 : 18

1, 6, 12 : 19

1, 4, 18 : 23

8, 3, 3 : 14

12, 2, 3 : 17

18, 2, 2 : 22

3, 4, 6 : 13

2, 4, 9 : 15

2, 6, 6 : 14

We can actually eliminate most of these now. They can't be 1, 8, and 9, because those add up to 18, and, significantly, that's the *only* combination that adds up to 18. Remember the guy knows the sum total (even if we don't). If it had been 18, he'd have worked out from that information alone that the ages were 1, 8, and 9.

You can eliminate all the combinations with a unique sum. If those had been the ages, the guy would not have needed that last clue.

In fact there are only two possible outcomes: 2, 6, and 6, or 8, 3, and 3. Of all the sets of 3 numbers less than 20 that have a product of 72, these are the only ones with a non-unique sum.

Had it been any of the other sets, the mathematician would have got the answer when he read the number of the sign.

So from this we can deduce that the number on the street sign must have been 14. We were able to deduce this from the fact that the mathematician was unable to work out the answer at this stage.

This is why your answer has to be wrong by the way. It can't be 2, 4, and 9, because that's the only set of numbers that add up to 15 that also have a product of 72. If those were the ages, then the street sign number would have been 15, and the mathematician would have worked out the answer as soon as he saw the number on the street sign.

So your answer has to be wrong.

Either that, or the mathematician wasn't very good. :)

The point about the house number is simply this:

We know the guy trying to work out the ages can see the number. We therefore know that *he* knows exactly what number it is. We don't know what the number is, but that doesn't matter. The important information revealed here is that the guy knows what the number is.

Why is that important information? Well, we know that even though he knows this number, that's insufficient information for him to work out the ages.

So what information do we have at this stage? We know the product of the ages is 72. And we know that even if you know the sum of the ages as well as the product, that's not enough to work out the individual ages.

I believe these are all the possible ages that give a product of 72 where the numbers are all under 20, and I've shown the sum of the ages next to them:

1, 8, 9 : 18

1, 6, 12 : 19

1, 4, 18 : 23

8, 3, 3 : 14

12, 2, 3 : 17

18, 2, 2 : 22

3, 4, 6 : 13

2, 4, 9 : 15

2, 6, 6 : 14

We can actually eliminate most of these now. They can't be 1, 8, and 9, because those add up to 18, and, significantly, that's the *only* combination that adds up to 18. Remember the guy knows the sum total (even if we don't). If it had been 18, he'd have worked out from that information alone that the ages were 1, 8, and 9.

You can eliminate all the combinations with a unique sum. If those had been the ages, the guy would not have needed that last clue.

In fact there are only two possible outcomes: 2, 6, and 6, or 8, 3, and 3. Of all the sets of 3 numbers less than 20 that have a product of 72, these are the only ones with a non-unique sum.

Had it been any of the other sets, the mathematician would have got the answer when he read the number of the sign.

So from this we can deduce that the number on the street sign must have been 14. We were able to deduce this from the fact that the mathematician was unable to work out the answer at this stage.

This is why your answer has to be wrong by the way. It can't be 2, 4, and 9, because that's the only set of numbers that add up to 15 that also have a product of 72. If those were the ages, then the street sign number would have been 15, and the mathematician would have worked out the answer as soon as he saw the number on the street sign.

So your answer has to be wrong.

Either that, or the mathematician wasn't very good. :)

# **Michael S. Kaplan** on 7 Jul 2006 8:23 AM:

Hi vdk2006,

I get it. That is not the problem.

But if I were giving the interview, I would tend to be more impressed by the person who can think beyond the numbers? :-)

I get it. That is not the problem.

But if I were giving the interview, I would tend to be more impressed by the person who can think beyond the numbers? :-)

# **Michael S. Kaplan** on 7 Jul 2006 8:28 AM:

Hi Ian,

As I said, I understand the answer as it is given. I was simply finding something that was more than numbers to be a bit more engaging?

Of course, I never went to M.I.T.

As I mentioned, I find myself most sympathetic to Jason's response:

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

*Hopefully old enough to run away from their freakish parents. I have to say, this test is starting to get a little weird. *

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

:-)

As I said, I understand the answer as it is given. I was simply finding something that was more than numbers to be a bit more engaging?

Of course, I never went to M.I.T.

As I mentioned, I find myself most sympathetic to Jason's response:

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

:-)

# **Joe** on 7 Jul 2006 10:16 AM:

Of course, it was silly of me to presume that you didn't do the probelm wrong on purpose to get at 'another way' at the problem. (How's that for a double negative?).

The purpose of these interview questions is to determine someone's problem solving ability. Unfortunately, this question is only targeting logical problem solving.

There are all sorts of problems to solve working on a software development team, and only some of them are logical in nature. Logical problem solving ability is really the bare minimum.

The purpose of these interview questions is to determine someone's problem solving ability. Unfortunately, this question is only targeting logical problem solving.

There are all sorts of problems to solve working on a software development team, and only some of them are logical in nature. Logical problem solving ability is really the bare minimum.

# **Michael S. Kaplan** on 7 Jul 2006 1:11 PM:

It would have been nice to see if the only factors that were not 100% numeric could also have some meaning here, but vdk2006's comment:

* BTW, I've seen this riddle in another language, and of course without any mention of Fairway or upper west side whatsoever... *

makes that unlikely. This is one of the reason I also used to hate word problems in math -- where every humanizing factor is just window dressing or fluff, and a distraction from the "real" problem. How often is that really the case in software?

Also, note that Bruce's earlier comment:

In response to Joe's claim that "The answer is 8,3,3 because it's unlikely that someone would refer to an older twin as 'my oldest.'":

*Two siblings of the same age need not be twins; they can be born, say, ten months apart. If it were in the window after the birthday of the younger of the two siblings but before the birthday of the next, they would be the same age (in whole years). *

Is indicative of the humanization "flaw" that the problem had -- it attempted to map to real life and manage to fail in that respect since it does not really map completely to something real....

makes that unlikely. This is one of the reason I also used to hate word problems in math -- where every humanizing factor is just window dressing or fluff, and a distraction from the "real" problem. How often is that really the case in software?

Also, note that Bruce's earlier comment:

In response to Joe's claim that "The answer is 8,3,3 because it's unlikely that someone would refer to an older twin as 'my oldest.'":

Is indicative of the humanization "flaw" that the problem had -- it attempted to map to real life and manage to fail in that respect since it does not really map completely to something real....

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