Read the given information carefully and answer the related questions:

Seven persons K, L, M, N, O, P and Q - were born (not in given order) on same date of same month but in different year – 1955, 1961, 1972, 1987, 1998, 2006, 2017. Consider base year 2025 for age calculation.

M’s age is the cube of a number. 11 years difference between M and K. Three persons are in between K and P. N is 32 years older to Q. L’s age is the cube of a number. Age difference between L and K is more than 10 years.

Four of the following five are similar in a certain way and form a group, who among the following is not related to the group?

Correct Answer: L

To solve this puzzle, let's calculate the age of each person in the base year 2025 using their birth years:

• Birth Year 1955: 2025 - 1955 = 70 years
• Birth Year 1961: 2025 - 1961 = 64 years
• Birth Year 1972: 2025 - 1972 = 53 years
• Birth Year 1987: 2025 - 1987 = 38 years
• Birth Year 1998: 2025 - 1998 = 27 years
• Birth Year 2006: 2025 - 2006 = 19 years
• Birth Year 2017: 2025 - 2017 = 8 years

Thus, the set of ages is: {70, 64, 53, 38, 27, 19, 8}.

Step 1: Identify the cubes of numbers among these ages
The ages that are perfect cubes are:
$4^3=64$
$3^3=27$
$2^3=8$
Since M's age and L's age are both perfect cubes, M and L must occupy two of the values from {64, 27, 8}.

Step 2: Use the difference between M and K
There is an 11-year difference between M and K. Let us test the possibilities for M:

• If M = 64: K can be $64-11=53$ (which is in the age list).
• If M = 27: K can be $27+11=38$ (which is in the age list).
• If M = 8: K can be $8+11=19$ (which is in the age list).

Step 3: Place P based on the position of K
The statement "Three persons are in between K and P" refers to their birth order (or ordered ages). Let's analyze the ordered positions:
Position order: 1st (70), 2nd (64), 3rd (53), 4th (38), 5th (27), 6th (19), 7th (8).
With three persons in between, the positions of K and P must be (1st, 5th), (2nd, 6th), or (3rd, 7th).

• If K is 3rd (53) and P is 7th (8): Then M must be 64 (since $64-53=11$). This leaves L to be the only remaining cube: L = 27. The difference between L and K is $53-27=26$, which is more than 10 years, satisfying the conditions.
• If K is 6th (19) and P is 2nd (64): Then M must be 8 (since $19-8=11$). This leaves L to be the remaining cube: L = 27. The difference between L and K is $27-19=8$ years, which is NOT more than 10 years. So this case is eliminated.

Thus, the correct ages are: K = 53, P = 8, M = 64, L = 27.

Step 4: Place N and Q
We are told N is 32 years older than Q. From the remaining ages {70, 38, 19}:
$70-38=32$
So, N = 70 and Q = 38. The remaining person O must be 19 years old.

Step 5: Summarize the final age table

Step 6: Find the odd one out among N, M, Q, and L

• N's age is 70 (an even number).
• M's age is 64 (an even number).
• Q's age is 38 (an even number).
• L's age is 27 (an odd number).

Similarly, N (1955), M (1961), and Q (1987) were born in odd years, whereas L (1998) was born in an even year.
Thus, L does not belong to the group.