Age Puzzle: Family Members

This problem requires us to determine the specific age of family member A by carefully analyzing the given information about seven family members (A, B, C, D, E, F, G). The members are ranked by age from oldest to youngest, with a consistent age difference between consecutive individuals.

Members' Ages Calculation

We are told that the seven family members are arranged by age, and the age difference between consecutive people is exactly 3 years. This setup indicates an arithmetic progression for their ages.

Let the ages from oldest to youngest be represented as $a_1, a_2, a_3, a_4, a_5, a_6, a_7$. The common difference ($d$) is 3.

We know the youngest person's age is 46. Therefore, $a_7 = 46$.

The formula for the $n$-th term in an arithmetic sequence is $a_n = a_1 + (n-1)d$. Since ages are decreasing, we can think of it as $a_n = a_1 - (n-1) \times 3$ or use $d=-3$. Using $a_7 = 46$ and $n=7$ with the common difference of 3:

$$a_7 = a_1 - (7-1) \times 3$$

Substituting the known value for $a_7$:

$$46 = a_1 - 6 \times 3$$

$$46 = a_1 - 18$$

Solving for $a_1$ (the age of the oldest person):

$$a_1 = 46 + 18$$

$$a_1 = 64$$

With the oldest person being 64 and the common difference being 3, we can list all the ages:

A's Age Clues Deduction

Now, let's apply the specific clues to identify A's age:

• B's Age: B is 61 years old. This places B as the second oldest person.
• B vs A: B is older than A. Since B is 61, A must be younger than 61. This narrows down A's possible ages to {58, 55, 52, 49, 46}.
• A's Age Restriction: A's age is not 55. This eliminates 55 from A's possibilities, leaving {58, 52, 49, 46}.
• A's Relation to D & E: A is older than both E and D. This means A cannot be the youngest person (46), as there would be no one younger than A. So, A's possible ages are now {58, 52, 49}.
• D's Age Restriction: D's age is not 55.
• E vs D: E is younger than D ($E < D$).
• G's Age Restriction: G's age is less than 52. From the calculated ages, G must be either 49 or 46.

A's Age Possibilities

Let's examine the remaining possibilities for A's age:

Possibility 1: A is 49

If A is 49, then D and E must both be younger than 49. The only available age younger than 49 is 46. Since D and E are different people, they cannot both be 46. Thus, A cannot be 49.

Possibility 2: A is 52

If A is 52, then D and E must be younger than 52. The available ages younger than 52 are 49 and 46.

Given $E < D$, D must be 49 and E must be 46.

Let's check the other conditions:

• B (61) is older than A (52) - True.
• A (52) is not 55 - True.
• A (52) is older than D (49) - True.
• A (52) is older than E (46) - True.
• D (49) is not 55 - True.
• E (46) is younger than D (49) - True.
• G's age must be less than 52. The remaining ages are {64, 58, 55}. None of these ages are less than 52. This contradicts the condition for G. Therefore, A cannot be 52.

Possibility 3: A is 58

If A is 58, then D and E must be younger than 58. The available ages younger than 58 are {55, 52, 49, 46}.

We know D's age is not 55. So, D's possible ages are {52, 49, 46}.

We also know $E < D$. Let's test the possibilities for D:

• If D is 52: E must be younger than 52. Possible ages for E are {49, 46}.
  • If E is 49: The ages assigned are B=61, A=58, D=52, E=49. The remaining ages are {64, 55, 46}. G must be younger than 52. G can be 46. This fits all conditions. The remaining ages {64, 55} can be assigned to C and F.
  • If E is 46: The ages assigned are B=61, A=58, D=52, E=46. The remaining ages are {64, 55, 49}. G must be younger than 52. G can be 49. This also fits all conditions. The remaining ages {64, 55} can be assigned to C and F.
• If D is 49: E must be younger than 49. The only available age is 46. So E must be 46. The ages assigned are B=61, A=58, D=49, E=46. The remaining ages are {64, 55, 52}. G must be younger than 52. Possible ages for G are 49 or 46. However, D and E are already assigned these ages. Therefore, D cannot be 49.
• If D is 46: E must be younger than 46. There are no ages available younger than 46. Therefore, D cannot be 46.

The only scenario that satisfies all conditions is when A's age is 58.

Final Answer: A's Age

After evaluating all the given conditions and possibilities, the age of A is determined to be 58 years.