In the pattern b w _ y z a, d s t u v _, _ o p q r e, choose the group of letters that should fill the blanks so that all three groups follow a logical pattern.

Introduction / Context:
This problem deals with three groups of six letters each, where three positions are missing. The groups are aligned so that comparing letters at the same position across groups reveals a systematic pattern. Our task is to find which triple of letters completes the pattern consistently in all six positions.

Given Data / Assumptions:

• Groups (with blanks): b w _ y z a, d s t u v _, _ o p q r e.
• There are three blanks: third in the first group, sixth in the second group and first in the third group.
• We expect an alphabetic relationship when letters from the same column are compared across the three groups.

Concept / Approach:
We arrange the three groups vertically by columns. For each column, we check whether there is a regular pattern such as constant steps forward or backward in the alphabet. Once we deduce the pattern for columns that are already complete, we apply it to the columns that contain blanks to find the missing letters.

Step-by-Step Solution:
Step 1: Write the groups in columns:Column 1: b, d, _Column 2: w, s, oColumn 3: _, t, pColumn 4: y, u, qColumn 5: z, v, rColumn 6: a, _, eStep 2: Examine Column 2: w (23), s (19), o (15). Each step is minus 4. Column 4 and Column 5 show the same pattern: y (25), u (21), q (17) and z (26), v (22), r (18) also decrease by 4 each step.Step 3: Apply this to Column 3. We have t (20) and p (16). Working backwards by 4 gives 24, which corresponds to x. So the first blank is x.Step 4: For Column 1, b (2) and d (4) increase by 2, so the third value should be 6, which is f. Thus the third group starts with f.Step 5: For Column 6, a (1) and e (5) suggest an arithmetic pattern with step 2, so the middle element must be 3, which is c.Step 6: The three blanks are therefore x (third in group one), c (sixth in group two) and f (first in group three).

Verification / Alternative check:
Rewriting all three groups with the filled blanks gives: b w x y z a, d s t u v c, f o p q r e. Checking every column now shows consistent arithmetic progressions in positions in the alphabet. No contradictions appear when we test all six columns together.

Why Other Options Are Wrong:
Other suggested triples, such as x, e, d or x, c, d, fail to preserve the strict arithmetic sequences within the columns, either breaking the constant difference of minus 4 or the plus 2 and plus 2 relationships. Only x, c, f satisfies all column wise patterns simultaneously.

Common Pitfalls:
Some learners only check one or two columns and neglect to verify the pattern across all six, leading to choices that satisfy partial constraints but not the full structure. It is important to ensure that a candidate triple works consistently in every relevant column.

Final Answer:
The letters that correctly fill the blanks are x, c, f in order.