Karnaugh Maps#
A Karnaugh map is a truth table rearranged so that simplification becomes something the eye can do. The rows and columns are ordered in Gray code — each cell differs from its neighbors by exactly one variable — so that physically adjacent 1s can always be combined into a simpler term. It turns the algebra of minimization into pattern-spotting.
Reading the Grid#
Here is a three-variable function, F(A, B, C), with the columns in Gray order (00, 01, 11, 10):
| A \ BC | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 | 0 |
The four 1s form a solid 2×2 block in the middle two columns. Those columns are exactly the ones where C = 1, and A and B take every value across the block — so A and B do not matter, and the whole function simplifies to just F = C. The canonical sum-of-products would have listed four minterms; the map shows at a glance that they collapse to a single literal.
The Rules of Grouping#
- Group adjacent 1s in rectangles whose size is a power of two — 1, 2, 4, 8. Bigger groups eliminate more variables, so make each group as large as possible.
- Adjacency wraps around the edges of the map, top-to-bottom and left-to-right, because the Gray ordering is circular.
- Don’t-cares may be included in a group when doing so enlarges it, and left out otherwise — free simplification from honestly recorded don’t-cares.
- Each group becomes one product term naming only the variables that stay constant across it; OR the group terms together for the minimized result.
When to Use One#
Karnaugh maps are practical up to about five or six variables; beyond that the adjacencies become impossible to see and algorithmic methods (Quine–McCluskey) or a synthesis tool take over. In modern practice a designer rarely draws one — the tools minimize automatically, and an FPGA lookup table stores the truth table directly, so shaving the last gate matters less than it once did. But the K-map remains the fastest way to reason about small logic by hand, and it builds the intuition for why a function simplifies — including, as the next page shows, where a correct-looking circuit can still glitch.