Karnaugh map reduction – when four adjacent 1s are grouped together, how many variables are eliminated from the final simplified expression?

Introduction / Context: Karnaugh maps (K-maps) are a visual technique to simplify Boolean expressions by grouping adjacent 1s into power-of-two blocks. This question checks whether you understand how grouping size relates to the number of variables eliminated from the resulting product term.

Given Data / Assumptions:

• We are working with a standard K-map (typically 2, 3, or 4 variables).
• Groups must be powers of two: 1, 2, 4, 8, … cells.
• Cells grouped together differ in only the variables that change across those cells.

Concept / Approach: Each time you double the group size, you eliminate one variable from the product term. A group of 1 eliminates 0 variables, a group of 2 eliminates 1 variable, a group of 4 eliminates 2 variables, and so on. Therefore, a group of four 1s will drop exactly two variables from the implicant because two coordinates vary over the grouping while the remaining coordinates stay constant.

Step-by-Step Solution:

Verification / Alternative check: Consider a 4-variable map (A,B,C,D). A 2x2 block spanning adjacent cells may vary in B and C while keeping A and D constant; the simplified term includes only the constant literals (e.g., A * D) and excludes B and C—confirming two variables are removed.

Why Other Options Are Wrong: “1” undercounts the effect for four cells; “3” overcounts because 4 cells cannot eliminate three variables unless the block were size 8; “4” confuses group size with variables removed; “Depends on adjacency” is incorrect because any valid 4-cell block is arranged so exactly two variables vary.

Common Pitfalls: Confusing cell count with variable count, and forgetting that only power-of-two rectangles are valid. Another pitfall is trying to eliminate variables that do not actually change across the chosen block.

Final Answer: 2