K-map grouping sizes: When grouping adjacent 1-cells in a Karnaugh map (for SOP), valid group sizes must be ________.

Introduction / Context:
Karnaugh map groupings represent implicants whose sizes determine how many variables drop out of the resulting product term. Only specific group sizes are allowed to preserve adjacency rules derived from Gray-code ordering.

Given Data / Assumptions:

• We are forming groups of adjacent 1s for SOP minimization.
• Adjacency includes edge wrap-around.
• Groups must be rectangular and uniform in size.

Concept / Approach:
Valid K-map group sizes are powers of two: 1, 2, 4, 8, 16, … This ensures every cell in a group differs from its neighbor by only one variable per step, enabling variable elimination in the product term. Arbitrary sizes like 3 or 5 break this property and are invalid.

Step-by-Step Solution:

Verification / Alternative check:
Translate a 2x2 group (size 4) into its product term and verify that two variables drop out; for a size-8 group, three variables drop out, confirming the power-of-two rule.

Why Other Options Are Wrong:

• 2's / 4's: These are valid sizes but incomplete; not the full allowed set.
• 3's: Not a power of two; invalid.
• Any size up to map width: Violates adjacency/literal-elimination properties.

Common Pitfalls:
Forgetting that singletons (size 1) are allowed; refusing to overlap groups when it yields larger power-of-two coverage; overlooking wrap-around adjacencies that enable size-4 or size-8 groups.

Final Answer:
1, 2, 4, 8, etc.