Solving the Letter-Number Series Puzzle

This question requires us to identify the logical pattern governing the series DEW33, IJB46, NOG59, STL72, ? and determine the missing cluster.

We can break down the series into three parts: the first letter, the second letter, the third letter, and the number. Let's analyze each part separately.

Analyzing the First Letter Pattern

Consider the first letter of each cluster: D, I, N, S.

• D is the 4th letter of the alphabet.
• I is the 9th letter of the alphabet.
• N is the 14th letter of the alphabet.
• S is the 19th letter of the alphabet.

The sequence of positions is 4, 9, 14, 19. The difference between consecutive terms is +5 ($9 - 4 = 5$, $14 - 9 = 5$, $19 - 14 = 5$).

Following this pattern, the next position should be $19 + 5 = 24$. The 24th letter of the alphabet is X.

Analyzing the Second Letter Pattern

Consider the second letter of each cluster: E, J, O, T.

• E is the 5th letter of the alphabet.
• J is the 10th letter of the alphabet.
• O is the 15th letter of the alphabet.
• T is the 20th letter of the alphabet.

The sequence of positions is 5, 10, 15, 20. The difference between consecutive terms is +5 ($10 - 5 = 5$, $15 - 10 = 5$, $20 - 15 = 5$).

Following this pattern, the next position should be $20 + 5 = 25$. The 25th letter of the alphabet is Y.

Analyzing the Third Letter Pattern

Consider the third letter of each cluster: W, B, G, L.

• W is the 23rd letter of the alphabet.
• B is the 2nd letter of the alphabet.
• G is the 7th letter of the alphabet.
• L is the 12th letter of the alphabet.

The difference between consecutive positions needs to account for wrapping around the alphabet (Z is followed by A). Let's calculate the difference:

• W (23) to B (2): The difference is 5 steps (W → X → Y → Z → A → B). Mathematically, this can be represented as $(2 + 26) - 23 = 5$, or using modulo arithmetic: $23 + 5 = 28$; $28 \pmod{26} = 2$.
• B (2) to G (7): The difference is $7 - 2 = 5$.
• G (7) to L (12): The difference is $12 - 7 = 5$.

The pattern is adding 5 positions each time, with wrap-around.

Following this pattern, the next position should be $12 + 5 = 17$. The 17th letter of the alphabet is Q.

Analyzing the Number Pattern

Consider the number in each cluster: 33, 46, 59, 72.

• $46 - 33 = 13$
• $59 - 46 = 13$
• $72 - 59 = 13$

The difference between consecutive numbers is consistently 13.

Following this pattern, the next number should be $72 + 13 = 85$.

Combining the Patterns

By combining the results from our analysis:

• The first letter is X.
• The second letter is Y.
• The third letter is Q.
• The number is 85.

Therefore, the missing letter-number cluster is XYQ85.