Read the given information carefully and answer the related question: Seven…

Concept

In a year-based age arrangement, convert each birth year to an age using age = base year - birth year, then place everyone on a single oldest-to-youngest timeline. A clue of the form "X persons were born between A and B" fixes the gap between two people; a clue that equates "persons before one person" with "persons after another" makes those two positions symmetric about the centre. No single clue usually fixes a position outright - you keep every layout each clue allows and intersect them until exactly one survives.

Set up the ages

With base year 2024, the seven birth years give these ages, listed oldest to youngest:

• 1955 -> 69, 1967 -> 57, 1973 -> 51, 1980 -> 44

• 1987 -> 37, 1991 -> 33, 2003 -> 21

Label the seven timeline slots 1 (oldest, age 69) to 7 (youngest, age 21).

Apply the clues

1. Balance clue: persons before Q = persons after P. If Q sits at slot q and P at slot p (counting from the oldest), this says q - 1 = 7 - p, i.e. p + q = 8. By itself this is only a symmetry condition; it permits several slot pairs such as (P,Q) = (1,7), (2,6), (3,5) and their mirrors - it does not yet fix who is oldest.

2. Add the gap clue: three persons between L and Q. L and Q must be four slots apart, so |slot(L) - slot(Q)| = 4.

3. Add L's age = multiple of 3. The multiples of 3 are 69, 57, 51, 33, 21 - i.e. slots 1, 2, 3, 6, 7. Intersecting the balance pairs, the four-slot L-Q gap, and L on a multiple-of-3 slot leaves only three candidate layouts: P at slot 1 / Q at slot 7 / L at slot 3; P at slot 3 / Q at slot 5 / L at slot 1; or P at slot 5 / Q at slot 3 / L at slot 7.

4. Use |age of Q - age of K| = 12 to break the tie. In the second layout Q would be 37 and in the third Q would be 51; neither leaves any remaining person whose age differs from Q by exactly 12. Only the first layout works: there Q is 21, and 21 + 12 = 33 is still available, so K = 33 (born 1991). This is what forces P to the oldest slot (69, born 1955) and Q to the youngest (21, born 2003).

5. Place L. L sits at slot 3, giving L = 51 (born 1973), which is indeed a multiple of 3.

6. Finish with M is 7 older than N, and N younger than O. The ages still unassigned are 57, 44 and 37 for M, N, O. Only 44 = 37 + 7 fits "M = N + 7", so M = 44 (born 1980) and N = 37 (born 1987); O then takes 57 (born 1967), and N < O holds.

Result

The final timeline (oldest to youngest) is P(69), O(57), L(51), M(44), N(37), K(33), Q(21). The required age difference is M - Q = 44 - 21 = 23 years.

Cross-check

Re-test every clue on this arrangement: L = 51 is a multiple of 3; exactly three persons (M, N, K) lie between L and Q; the count older than Q (six) equals the count younger than P (six); |Q - K| = |21 - 33| = 12; M = 44 = 37 + 7 = N + 7; and N = 37 < 57 = O. Every clue holds, so 23 years is confirmed.