Me and my spouse have a big family, comprising 8 females and 6 males. We have two sons and two daughters, of whom the eldest is unmarried. Out of my two grandsons and three granddaughters, only one grandson is a single child. His mother is my younger daughter. Which of my children necessarily has two daughters?
Family Structure Analysis
The speaker has four children: two sons (let's denote them S1, S2) and two daughters (Elder Daughter ED, Younger Daughter YD).
The total number of grandchildren considered is 2 grandsons and 3 granddaughters.
Grandchildren Parental Analysis
The statement "Out of my two grandsons and three granddaughters, only one grandson is a single child. His mother is my younger daughter" provides key information:
- The Younger Daughter (YD) has exactly one child, who is a grandson (GSm1).
- Therefore, YD has 0 daughters.
- The remaining grandchildren are 1 grandson (GSm2) and 3 granddaughters (GD1, GD2, GD3).
- These remaining grandchildren must be the offspring of the other children: S1, S2, and ED.
Impact of Eldest Child Constraint
The condition "of whom the eldest is unmarried" is crucial. We analyze based on who the eldest child could be:
Scenario 1: The Elder Daughter (ED) is the eldest child.
- If ED is the eldest, and she is unmarried, she has no children (0 daughters).
- Consequently, all 3 granddaughters (GD1, GD2, GD3) must be daughters of the two sons (S1 and S2).
- Let $N_{GD(S1)}$ be the number of daughters of S1 and $N_{GD(S2)}$ be the number of daughters of S2. The total number of granddaughters from them must sum to 3:
$ N_{GD(S1)} + N_{GD(S2)} = 3 $
- Given that $N_{GD(S1)}$ and $N_{GD(S2)}$ are non-negative integers, this equation necessitates that at least one of them must be 2 or 3. (Possible pairs (S1, S2): (0,3), (1,2), (2,1), (3,0)).
- Therefore, in this scenario, it is necessary that one of the sons has at least two daughters.
Scenario 2: An Elder Son (S1 or S2) is the eldest child.
- If an eldest son (e.g., S1) is unmarried, he has 0 children.
- The 3 granddaughters must then be distributed between the other son (S2) and the Elder Daughter (ED). The equation becomes $N_{GD(S2)} + N_{GD(ED)} = 3$.
- In this case, it's possible that $N_{GD(S2)} = 1$ and $N_{GD(ED)} = 2$. Here, the Elder Daughter has two daughters, but S2 does not necessarily have two daughters. This contradicts the requirement that the condition must *necessarily* hold.
Conclusion on Necessary Condition
To satisfy the "necessarily" requirement of the question, Scenario 1 must be assumed as the intended interpretation. The family structure implies the Elder Daughter is the eldest and unmarried, forcing the two sons to be the fathers of all three granddaughters. This guarantees that one son must have two daughters.
Important Questions from Blood Relations
- Consider the relationships among P, Q, R, S, and T:
• P is the brother of Q.
• S is the daughter of Q.
• T is the sister of S.
• R is the mother of Q.
The following statements are made based on the relationships given above.
(1) R is the grandmother of S.
(2) P is the uncle of S and T.
(3) R has only one son.
(4) Q has only one daughter.
Which one of the following options is correct? Given the statements:
- P is the sister of Q.
- Q is the husband of R.
- R is the mother of S.
- T is the husband of P.
Based on the above information, T is _______________ of S.
- M and N had four children P, Q, R and S. Of them, only P and R were married. They had children X and Y respectively. If Y is a legitimate child of W, which one of the following statements is necessarily FALSE?
- Each of P, Q, R, S, W, X, Y and Z has been married at most once. X and Y are married and have two children P and Q. Z is the grandfather of the daughter S of P. Further, Z and W are married and are parents of R. Which one of the following must necessarily be FALSE?
M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in- law of M. How is P related to M?
