Karnaugh map (K-map) grouping rules — which is invalid? In K-map simplification, which of the following cell combinations cannot be grouped as valid adjacencies to form implicants?

Introduction / Context:
Karnaugh maps exploit adjacency of minterms or maxterms to eliminate variables and simplify Boolean expressions. Understanding valid adjacencies (including wrap-around at edges) prevents mistakes that lead to incorrect simplifications or missed opportunities for larger groups.

Given Data / Assumptions:

• Standard Gray-code ordering is used on K-map rows/columns.
• Adjacency exists between cells that differ in exactly one variable.
• Edge wrap-around connects left-right edges and top-bottom edges.

Concept / Approach:
Valid K-map groups are rectangles with areas that are powers of two (1, 2, 4, 8, ...). Cells on opposite edges are adjacent due to wrap-around. However, diagonal cells differ in two variables and are not adjacent, so they cannot be grouped directly without intermediate cells.

Step-by-Step Solution:

Verification / Alternative check:
Write Gray-coded labels and compute Hamming distance between diagonal cells; distance = 2 → no adjacency.

Why Other Options Are Wrong:

• Row/column corners: legal due to wrap-around.
• Overlapping: legal and often useful for creating larger implicants.

Common Pitfalls:
Forgetting wrap-around adjacency or forcing diagonal groupings that skip required intermediate cells, leading to incorrect minimized expressions.

Final Answer:
Diagonal corners