Logical Reasoning: Solving the Letter-Number Series Puzzle

This solution explains how to find the missing element in the given series: DEW30 JKC45 PQI60 VWO75 ?. We will break down the series into its components: the first letters, the second letters, the third letters, and the numbers, and identify the pattern for each.

Pattern Analysis

1. Number Series Pattern

Observe the numbers in the series: 30, 45, 60, 75.

• The difference between 45 and 30 is $45 - 30 = 15$.
• The difference between 60 and 45 is $60 - 45 = 15$.
• The difference between 75 and 60 is $75 - 60 = 15$.

The pattern shows a consistent addition of 15 to the previous number. Therefore, the next number in the series will be $75 + 15 = 90$.

2. First Letter Series Pattern

Consider the first letters of each cluster: D, J, P, V.

Let's find their positions in the English alphabet (A=1, B=2, ... Z=26):

Now, let's look at the difference between consecutive positions:

• $10 - 4 = 6$
• $16 - 10 = 6$
• $22 - 16 = 6$

The pattern is adding 6 to the position of the previous letter. The next position is $22 + 6 = 28$. Since there are only 26 letters in the alphabet, we find the corresponding letter by calculating $28 \pmod{26}$. A position of 28 is equivalent to the $28 - 26 = 2$nd position, which is the letter B.

3. Second Letter Series Pattern

Consider the second letters of each cluster: E, K, Q, W.

Find their positions in the alphabet:

Analyze the difference between consecutive positions:

• $11 - 5 = 6$
• $17 - 11 = 6$
• $23 - 17 = 6$

The pattern is again adding 6 to the position of the previous letter. The next position is $23 + 6 = 29$. Using modulo arithmetic, the 29th position corresponds to the $29 - 26 = 3$rd position in the alphabet, which is the letter C.

4. Third Letter Series Pattern

Look at the third letters: W, C, I, O.

Find their positions in the alphabet:

Let's examine the progression, considering alphabet wrap-around (modulo 26):

• From W (23) to C (3): The step is $23 + 6 = 29$. Since $29 \equiv 3 \pmod{26}$, the difference is +6.
• From C (3) to I (9): The step is $3 + 6 = 9$. The difference is +6.
• From I (9) to O (15): The step is $9 + 6 = 15$. The difference is +6.

The pattern is consistently adding 6. The next position is $15 + 6 = 21$. The 21st letter of the alphabet is U.

Conclusion

Combining the results from the four patterns:

• The next number is 90.
• The next first letter is B.
• The next second letter is C.
• The next third letter is U.

Therefore, the complete cluster that replaces the question mark is BCU90.