The Missing House Number

Question

The Visit

One day I was visiting my sister in Boston and heard a bunch of kids yelling and playing in the yard ( yahd ).

I couldn't see the kids, and hadn't seen my sis for years so I asked her, "are all them kids yours, sis?"

"Hell no!", she laughed. "My kids are playing with their friends from three other families in the neighborhood, but ours is the biggest."

She explained to me that the Boyd's have a smaller number of kids, the Gilmore's have an even smaller number, and the Knight's have the smallest number of kids out of all of us.

"Interesting, how many kids all together?", I asked her.

"Let me sum it up like this, bro... there are fewer than 18 kids and the product of the numbers in the four families happens to be my house number which I texted to you for directions here."

Her house number...darn, phone's in the car in the text message.

I took out a piece of paper and started scribbling frantically.

"Sis, I need more information. Do the Knight's have more than 1 kid?"

My sister then told me,

"Bro actually you do not need any further info to figure this out, but I will tell you anyways."

As soon as she answered, I correctly told her the number of kids in each family, leaned over and hugged her goodbye, and left.

I figured this out based on the info provided, can you? Oh, you might need my sister's last name - Smith.

---

The Missing House Number What is it?

and...

How many kids are in each family?

Answer 1

I figured out the trick to this problem. The answer is

Missing house number is 108.
The Smith's have 9 kids, Boyd's have 4 kids, the Gilmore's have 3 kids, the Knight's have 1 kid.

The trick here is something that I can deduce about my sister. She said that

I do not need any further info to figure this out. And she also thinks that I know the house number. Obviously the house number cannot be factored into multiple sums below 18, such as 120, 80, 96 etc. because then I would need more info, so we can eliminate all the duplicates right off the bat.

But there is something more about my sister. If she knows that I know the house number, why did she specify the sum of the children is fewer than 18? If her house number was 105 for example, I can only factor that into 1, 3, 5, and 7 - no other factorizations are possible, so I would know the answer without having been told the sum of the children, just that the product is the house number. My sister doesn't do anything for no reason, so I'll assume the sum is important. Because of this I can eliminate all the house numbers that give me the answer unambiguously - such as 66 (1,2,3,11), 42 (1,2,3,7) etc. as well as house numbers that don't entirely consist of primes, such as 56 (1,2,4,7) or 54 (1,2,3,9) because these cannot be made into 2 separate sets of 4 distinct integers and thus will give away the answer.

What I am left with is only 8 choices

1 3 4 9
1 3 6 7
1 4 5 7
2 3 4 7
2 3 5 6
2 3 4 8
2 3 5 7
2 4 5 6
Now all of these have distinct products, and if I remembered the house number I would already know the answer, but I don't. However, again why did my sister mention the sum was fewer than 18? Well, because 1 3 6 7 has a product of 126, but so does 1 2 7 9. So if her house number was 126 then I wouldn't be able to tell if there were 17 or 19 kids. So, trying to be helpful, she mentioned that the sum is below 18. But wait, why didn't she say it was fewer than 19? Obviously because her actual house number has a duplicate in the 18 kid category, otherwise she would have said 19 just to make my life that much harder. Out of the 8 choices I have left, only 3 have duplicates with total sum of 18 kids:

1 3 4 9 = 108 = 1 2 6 9 (18)
2 3 4 7 = 168 = 1 4 6 7 (18)
2 4 5 6 = 240 = 2 3 5 8 (18)

As you can see

Two of the choices have Knight's having 2 kids and only 1 choice has them having 1 kid. My sister was totally right, if I had the house number I wouldn't need any more info, but as I forgot it I asked how many kids the Knight's have. Since she answered me and I was able to give her the answer, she must have said they only had 1 kid, eliminating every possibility but one. And that's the answer.

Answer 2

I'm going to take a stab here and say:

The Knights have one kid.
The Gilmores have three kids.
The Boyds have five kids.
The Smiths, your sister's family, have seven kids.

There doesn't seem to be a way to constrain the information provided in a way to give a single answer. There are many ways to sum four distinct numbers such that their total is less than 18, and their products all form numbers that would be perfectly plausible as house numbers.

We might somewhat arbitrarily decide a house number should be three digits, and that as is done in many places, the numbering for your sister's street uses the hundreds place (and greater, if any) as a block number and the tens/ones places as a house number within that block. We note too that most blocks don't have all that many houses, so let's also arbitrarily constrain the highest tens/ones value to, say, 20. That still leaves plausible results of:

1 3 4 9 = 108
1 3 5 7 = 105
1 3 5 8 = 120
1 3 6 7 = 126
1 4 5 6 = 120
2 3 4 5 = 120
2 3 5 7 = 210

Even knowing, say, that the Knights have more than one kid, still doesn't decide between 120 and 210.

However, your sister tells you you don't need to know how many kids the Knights have to figure this out. This suggests that there is something distinctive here; if, for example, we know that your sister lives on the side of the street with odd numbering, then only the 1*3*5*7=105 solution would apply. And since there's only one odd product out of all the possibilities for four numbers under 18, knowing that you can arrive at a distinct solution points us at that one distinct possibility. So I surmise that is the answer here.

Answer 3

The problem as stated does not match the problem that John S. described in the hint. She said "Bro actually you do not need any further info to figure this out, but I will tell you anyways.", but he does not have enough for the intended product. This can be fixed by changing the sister's comment to something like: "That's right. I didn't give you enough information, so I'll answer your question."

Also, it doesn't matter whether the narrator remembered the house number at the time the events took place.

The possible ways to full her initial comments are:
24 1 2 3 4
30 1 2 3 5
36 1 2 3 6
40 1 2 4 5
42 1 2 3 7
48 1 2 3 8
48 1 2 4 6
54 1 2 3 9
56 1 2 4 7
60 1 2 3 10
60 1 2 5 6
60 1 3 4 5
64 1 2 4 8
66 1 2 3 11
70 1 2 5 7
72 1 2 4 9
72 1 3 4 6
80 1 2 4 10
80 1 2 5 8
84 1 2 6 7
84 1 3 4 7
90 1 2 5 9
90 1 3 5 6
96 1 2 6 8
96 1 3 4 8
105 1 3 5 7
108 1 3 4 9
120 1 3 5 8
120 1 4 5 6
120 2 3 4 5
126 1 3 6 7
140 1 4 5 7
144 2 3 4 6
168 2 3 4 7
180 2 3 5 6
192 2 3 4 8
210 2 3 5 7
240 2 4 5 6

If her house number were 24, 30, 36, 40, 42, 54, 56, 64, 66, 70, 105, 108, 126, 140, 144, 168, 180, 192, 210, or 240, there is only one way to get that product; he could have solved it immediately if he remembered the number. Even if he didn't he could eliminate those possibilities after she said he didn't have enough information.

Eliminating those possibilities leaves:
48 1 2 3 8
48 1 2 4 6
60 1 2 3 10
60 1 2 5 6
60 1 3 4 5
72 1 2 4 9
72 1 3 4 6
80 1 2 4 10
80 1 2 5 8
84 1 2 6 7
84 1 3 4 7
90 1 2 5 9
90 1 3 5 6
96 1 2 6 8
96 1 3 4 8
120 1 3 5 8
120 1 4 5 6
120 2 3 4 5

If her number were any number other than 120, then learning how many children the Knights had would be useless. She could have even said, "You should be able to work that out without my telling you." If he knows her house number is 120, and is told that there is one Knight child, he is unable to work out how many children there are. But if he is told there are 2 Knight children, then he could deduce there must be 2 Knight, 3 Gilmore, 4 Boyd, and 5 Smith children.