Good girls and bad boys (10)

Suppose the youngest one is a girl. Then she indeed has three elder sisters such that all four are girls. This would mean four true statements. However, the statements made by the three older children are invalid. So we have a contradiction.

Hence the youngest must be a boy. Snce he is a liar, not all of his three elder siblings are girls - at least one is a boy. In total, at least two out of four are boys.

Suppose all four are boys. Then we have four lies. But the oldest sibling is not lying as he indeed has not more than one younger sister. So this case is impossible.

Now suppose now three are boys and one is a girl. Let the oldest sibling be the only girl. Yes, she is indeed telling the truth as she really has at most one younger sister. But the second child would be right in saying he has exactly two brothers (third and youngest child). As a boy, he is however supposed to lie. We have a contradiction.

Let the second sibling be the only girl. We also have a problem as she is supposed to be a truth teller by virtue of her gender. But she actually has three brothers (one older, two younger), not two. We also have a contradiction.

Let the third sibling be the only girl. Same thing here, as a girl, she is supposed to be a truth teller. However, she is lying by saying two are girls as she is the only girl. Another contradiction. We need not try out the case for the youngest child since we already proved that he must be a boy.

Hence it is impossible for there to be three boys and a girl.

Combining these with the fact that at least two are boys, we know the number of boys must be exactly two. Hence there are exactly two girls. Two possibilities are shown below:

Boy, girl, girl, boy. OR Girl, boy, girl, boy.

Note: Whichever way, the third child must be a girl since it is true that half are girls.

The intro says nothing about the 4 children being related to each other. All statements of the children talk about their respective siblings, which is inconsequential to the question, except one, who says that half of them are girls. Thus the answer is 2. But only if that child is one of the two girls.

Assuming all 4 to be siblings, the youngest said something that contradicts all the other statements. Thus, even if the youngest really have 3 elder sisters as claimed, then he must be their younger brother (believing the youngest means believing the supposed elder sisters, and all of them claim to have male siblings), and as such, a liar.

By knowing the youngest sibling is a liar, we know that at least one of the elders must have been a boy. Therefore, if the number of boys and girls are not equal, then there must be more boys than girls among them with a girl eldest sister, a boy third sibling and "unable to be determined" gender for second eldest. With this, we can see that boy > girl is impossible the same way girl > boy from the first paragraph.

If we analysed each of the elder siblings' statements separately, we can conclude that all of them (as an independent true statement each) are implying that the number of boys and girls are really equal.

• The third elder sibling's statement was from an outsider's POV : (half are girls) —X— (more boys or more girls)
• The second elder sibling's statement was from an insider's POV : (exactly two brothers) —X— (exactly 1 or 3 brothers)
• The eldest sibling's statement was from an insider's POV : (at most one sister) —X— (at least two sisters)

Looking at the table above, only 2 brothers and 2 sisters scenarios can be found in all of the elder siblings' possibilities (with 4 boys impossible for eldest nor 3 boys possible for second eldest). Therefore, the earlier inference from all three statements really came through.