Counting siblings problem

Question

A person has told us that at least half of his or her siblings have exactly three sisters and not more than half of his or her siblings have at least four brothers. How many arrangements of brothers and sisters matching this description are there?

The only arrangement I found is where the asked person is a boy and he has three sisters and three brothers. Is there a way to prove that it is the only one? Or are there any other arrangements?

EDIT: Okay so I found another option where the person is a girl with two sisters and four brothers.

Answer 1

=== new answer ===

Notice whoever has exactly $3$ sisters and who has at least $4$ brothers are gender delineated:

If there are more than $4$ and fewer than $3$ girls then no-one will have exactly $3$ sisters. This will mean that $0$ is at least half the siblings so there are no siblings and the speaker is an only child.

If there are $4$ girls then the girls have exactly $3$ sisters.

If there are $3$ girls then the boys have exactly $3$ sisters.

If there fewer than $4$ boys then no one has at least $4$ brothers.

If there are $4$ boys then the girls have at least $4$ brothers.

If there are more than $4$ boys everyone has at least $4$ brothers. This means that all siblings is at most half of the siblings. So there are $0$ siblings which is a contradiction to anyone having brothers. This is not an option.

So it's a matter of determining if the boys or the the girls are less, equal, or more than half the siblings.

Case 1: No-one has exactly $3$ sisters.

Speaker is only child.

Case 2: the girls have exactly $3$ sisters.

Than the sisters are equal or more than half and the brothers are equal or less than half. Only the girls can have at least $4$ brothers so either the girls are both at least and at most half (i.e. exactly half), or no-one has $4$ brothers.

Case 2a: the girls have exactly $3$ sisters and at least $4$ brothers.

Sister has equal numbers as brothers and sisters. There are $4$ girls and there are $4$ boys. So the speaker is genderless. Presumably this is not an option. If it were it really makes the problem unsolvable as any number of siblings could be neither brothers nor sisters.

Case 2b: the girls have exactly $3$ sisters and no one has $4$ brothers.

There are $4$ girls and less than $4$ boys, and the sisters (either $3$ or $4$) equal or outnumber the brothers.

$4$ girls and $0,1,2,3$ boys.

Case 3: the boys have exactly $3$ sisters.

Then there are $3$ girls and the brothers, outnumber or equal the sisters who are the only ones that can have $4$ brothers. If the speaker is a boy there are the $3$ sisters and at least $3$ brothers. (That is there are $4$ boys.) If the speaker is a girl there are are $2$ sisters and at least $2$ brothers.

$3$ girls, $2$ boys (speaker must be a girl)

$3$ girls, $4$ boys (speaker must be a boy).

===old answer ===

There are either $3$ or $4$ girls in the family. That is because at least half have exactly $3$ sisters. The boys will have $3$ sisters if there are $3$ girls, and the girls will have $3$ sisters if there are $4$ girls.

There are at most $4$ boys. That is because if there were $5$ or more everyone would have at least $4$ brothers.

Because the boys don't count themselves among the brothers, each boy has fewer brothers than the girls do. So if anyone actually has four brothers then it is the girls.

Case 1: There are $4$ boys so the girls have at least $4$ brothers. At most half the speakers siblings are girls. If the speaker is a boy, he has $3$ brothers and $3$ or fewer sisters. But he must have at least $3$ sisters or else no-one will have $3$ sisters. SO if her has $3$ sisters the his $3$ brothers will cclaim to have exactly $3$ sisters. That's exactly half

So $4$ boys, $3$ girls and the speaker is a boy is a possibility.

If the speaker is a girl, she has $4$ brothers. She has $2$ or $3$ sisters who claim to have at least $4$ brothers. That's less than half so that's okay. If there are $4$ girls then the $3$ sisters will claim to have $3$ sisters and that is less than half. So there are $3$ girls.

So $4$ boys, $3$ girls and the speaker is a girl is a possibility.

Case 2: There are fewer than $4$ boys. Then no-one claims to have at least $4$ brothers and that is certainly less than half.

If there are $4$ girls the speaker will have $3$ or $4$ sisters claiming to have exactly $3$ sisters. The speaker will have at most $3$ brothers so those claiming to have $3$ sisters will be at least half..

So $4$ girls and $0,1,2$ or $3$ boys and the speaker being a boy or a girl are a possibility.

If there are $3$ girls then it is the boys that will claim to have sisters. So the speaker will need to have at least as many brothers as sister. There are at most $3$ boys. If the speaker is a boy then he will have $3$ sisters but at most $2$ brothers so that is an impossibility.

If the speaker is a girl, then she has $2$ sisters and she can have $2$ or $3$ brothers.

So those are the last possibilities. $3$ girls and $2$ or $3$ boys and the speaker is a girl.

Answer 2

A child with three sisters can either be a girl in a family with four girls or a boy in a family with three girls. The part about brothers will cap the number of boys at four or five, but there is nothing to force there to be any boys at all.

If there are only three girls it is the boys who have three sisters. A girl speaking will have two siblings with two sisters, so must have at least two brothers and no more than four. A boy speaking will have three siblings with two sisters, so must have three or four brothers.

If there are four girls it is the girls that have three sisters. A girl speakng will have thee siblings with three sisters, so there can be at most three boys. A boy speaking will have four siblings with three sisters, so can afford to have up to three brothers.

This gives the following possibilities: $$\begin {array} {c c c c}\\ boys & girls & boy speaking & girl speaking\\ \hline 2&3&No&Yes\\ 3&3&No&Yes\\ 4&3&Yes&Yes\\5&3&Yes&No\\0&4&No&Yes \\1&4&Yes&Yes\\2&4&Yes&Yes\\3&4&Yes&Yes\\4&4&Yes&No \end {array}$$ There are nine different family structures that can meet the requirement. If we count separately cases where a boy or girl is speaking there are thirteen.

Answer 3

If a boy has 3 sisters then it will be 3 girls. If a girl has 3 sisters, then it will be 4 girls.

Consider each case.

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3 girls.

Therefore 1st statement refers to boys. If 2nd statement also refers to boys, then it should be no less than 5 boys, which always makes more than half siblings.

So, 2nd statement must refer to girls, and with 4 or more boys it will be true no matter who is the speaker.

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4 girls

1st statement refers to girls.

If a girl is speaking, then 1st statement can only be true with 3 or less boys. In this case the 2nd statement must refer to none of siblings, otherwise it will be wrong (anyone having less than 4 brothers).

A boy speaker must have at least 4 brothers (5 boys total) to have 1st statement true. If the 2nd statement refers to girls, both statements are correct with 5 or more boys. If 2nd statement refers to boys, there must be exactly 5 boys, otherwise the statement is wrong (over half siblings).

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Answer: 3 girls and 4+ boys, 4 girls and 0,1,2,3 or 5+ boys

PS. Thanks to Ross Millikan for useful comments.