Understanding the Family Age Puzzle

This problem requires us to determine the ages of seven family members (A, B, C, D, E, F, G) based on a set of clues. The family members are arranged by age from oldest to youngest, with a specific age difference between consecutive individuals. We need to find out which person's age is exactly 55.

Deducing the Age Sequence

The problem states that the family consists of seven members, and the age difference between consecutively arranged persons is 3. This indicates that their ages form an arithmetic progression. We are also given that the youngest person's age is 46.

Using the information that the youngest is 46 and the consecutive age difference is 3, we can list all seven ages:

• The youngest person's age is 46.
• The next oldest person's age is $46 + 3 = 49$.
• The next oldest is $49 + 3 = 52$.
• Continuing this pattern, the ages are: 46, 49, 52, 55, 58, 61, 64.

Arranging these ages from oldest to youngest, the sequence is: 64, 61, 58, 55, 52, 49, 46.

Placing Known Members and Applying Constraints

Now, we use the given clues to place the individuals within this age sequence:

• B's Age: B is 61 years old. This places B as the second oldest person.
• B and A Relationship: B is older than A. Since B is 61, A must be younger than 61.
• A's Age Restriction: A's age is not 55.
• A's Relation to D and E: A is older than both D and E.
• E and D Relationship: E is younger than D.
• D's Age Restriction: D's age is not 55.
• G's Age Restriction: G's age is less than 52.

Determining A's Age

From the age list (64, 61, 58, 55, 52, 49, 46), A must be younger than B (61) and A's age is not 55. So, A's possible ages are 58, 52, 49, or 46.

The condition that A is older than two distinct individuals (D and E) is crucial. If A were 49, only the age 46 is younger, which is not enough for two people. If A were 46, no one is younger. Therefore, A must be either 58 or 52.

Let's test the possibility of A being 52:

• If A is 52, then D and E must be younger than 52. The only available ages younger than 52 are 49 and 46.
• Given that E is younger than D, D must be 49 and E must be 46.
• The age sequence would be: ?, B(61), ?(C/F/G), A(52), D(49), E(46).
• The remaining people are C, F, and G. The remaining ages are 64, 58, and 55.
• We know that G's age must be less than 52. However, the ages less than 52 (49 and 46) are already assigned to D and E. This means G cannot be placed.

This contradiction shows that A cannot be 52. Thus, A's age must be 58.

Determining D's Age

With A being 58, the age sequence is: 64, 61(B), 58(A), ?, ?, ?, ?.

The remaining ages are 55, 52, 49, and 46 for the people C, D, E, F, and G.

We apply the remaining constraints:

• D's age is not 55.
• A (58) is older than D.
• E is younger than D.
• G's age is less than 52 (meaning G is 49 or 46).

Since D's age is not 55, and A (58) is older than D, D can potentially be 52, 49, or 46.

Let's examine these possibilities for D:

• If D = 49: E must be younger than 49, so E would be 46. This assigns ages 49 to D and 46 to E. The remaining ages less than 52 are none. G must be less than 52. However, ages 49 and 46 are already taken by D and E. Thus, G cannot be placed. This leads to a contradiction.
• If D = 46: E must be younger than 46. There is no age available younger than 46 in the list. This is also a contradiction.

Therefore, D must be 52 years old.

The age sequence is now partially filled as: 64, 61(B), 58(A), ?, 52(D), ?, ?.

Placing E and G

The remaining ages are 55, 49, and 46.

The remaining people are C, E, F, and G.

We know the following:

• E is younger than D (52), so E can be 49 or 46.
• G's age is less than 52, so G can be 49 or 46.

Since E and G must occupy two distinct ages, and the only available ages less than 52 are 49 and 46, these ages must belong to E and G (in any order).

This leaves the age 55 for one of the remaining people, which are C and F.

Identifying the Person Aged 55

Based on our deductions, the ages and assigned members are:

As deduced, the person with age 55 must be either C or F.

Now let's check the given options:

• 1. G: G's age is either 49 or 46.
• 2. F: F could be 55.
• 3. A: A's age is 58.
• 4. C: C could be 55.
• 5. Either 2 or 4: This option states that either F or C is 55.

Since the age 55 can belong to either F or C, the option "Either 2 or 4" correctly identifies the possibilities.