# How many children

A man invites his friend to his house for a small chatter over some tea. There is a good number of children playing in the backyard, and upon seeing this, the guest asks his host whether all these children are his. The host gives him a brief "no" and tells him that the children belong to himself and three other families. The guest redoubles curiosity and asks to his host how many children he has. The host answers him: "Let's put it this way: the total number of children playing in the background is less than 18, with each family having a different number of children, mine being the largest. Also, the product of the number of children in each family happens to be my home number which you just saw when you arrived". The guest then scrambles on piece of paper for a bit and then looks up and says "I need more information. Does the family with the smallest number of children have more than one child?". As soon as the host answers, the guest is able to correctly tell how many children the latter has. All of this having been said, anyone can do the same on the basis of the information given above. How many children does the man in question have?

Last edited:

How many children does the man in question have?
He has 5 children.

DaveC426913
Gold Member
I'm missing a piece. I don't see why he couldn't have 6.

I'm missing a piece. I don't see why he couldn't have 6.
And what would be the home number, in this case?

DaveC426913
Gold Member
And what would be the home number, in this case?
Well, I didn't really know what that meant. Address? Phone number? My phone number is ten digits.

Dave, it's important to remember the question the guest asks.

DaveC426913
Gold Member
Dave, it's important to remember the question the guest asks.
Yes, the way I see it, that eliminates 9 out of 12 possibilites, narrowing it down to 3.

It's definitly solvable

DaveC426913
Gold Member
Is this relevant? "my home number which you just saw when you arrived".
I don't see how that provides any information for filtering. At least not a universal one. Perhaps it's Yankee-centric?

Well, according to your calculations, how much is the product of the number of children in each family?

DaveC426913
Gold Member
Well, according to your calculations, how much is the product of the number of children in each family?

Either 120 or 180 or 240.

DaveC426913
Gold Member
Hm. No, I'm definitely doing something wrong. Upon review, my notes reveal a plethora of answers. I'm missing several pieces.

Either 120 or 180 or 240.
If it was 180, for instance, the numbers would have to be a combination of the factors
2*2*3*3*5.
Considering the possibility of a family with just 1 child, how many children would the other families have?

DaveC426913
Gold Member
If it was 180, for instance, the numbers would have to be a combination of the factors
2*2*3*3*5.
2,3,5,6 works.

2,3,5,6 works.
No, it doesn't. Answer the post #13, and you will see why it doesn't work.

DaveC426913
Gold Member
If family A had just one child, then the possibilities are virtually limitless.
1,2,3,4
1,2,3,12
1,4,5,7
or anything in between

DaveC426913
Gold Member
The question here isn't whether it's solvable, the question is whether it's solvable without certain culture-centric conventions (such as certain values for a "home number").

If family A had just one child, then the possibilities are virtually limitless.
1,2,3,4
1,2,3,12
1,4,5,7
or anything in between

If the product is 180, and one family has one children, how many children would the other families have?
For instance, 4,5,9 doesn't works, since the sum 1+4+5+9 > 18 ...

Dave, your not concluding the most out of this part: "As soon as the host answers, the guest is able to correctly tell how many children the latter has."

The question here isn't whether it's solvable, the question is whether it's solvable without certain culture-centric conventions (such as certain values for a "home number").
Forget the "home number". Consider it was just a number written in a piece of paper.

DaveC426913
Gold Member
Dave, your not concluding the most out of this part: "As soon as the host answers, the guest is able to correctly tell how many children the latter has."
Yeah, there're only two possibilities, either the smallest family has 1 or it has more. That means at least one of those conditions has only one solution. As soon as he knows it's that one, he knows the answer.

Dave, you do realize that the fact that the guest does know the home number is very important, right?

DaveC426913
Gold Member
Dave, you do realize that the fact that the guest does know the home number is very important, right?
No.
10 chars

Well, start realizing!

DaveC426913
Gold Member
I think I'd better leave this for those more skilled.