Karnaugh Maps
A Karnaugh map is a grid arrangement of a truth table that makes Boolean simplification a matter of grouping adjacent 1s.
What it is
A Karnaugh map, or K-map, is a truth table redrawn as a grid whose rows and columns are labeled in Gray-code order, so any two neighboring cells differ in exactly one input. Simplification becomes visual: group adjacent 1s and each group collapses to a smaller product term.
Gray-code adjacency
Cells are ordered 00, 01, 11, 10 rather than in binary counting order. This ensures single-variable changes between neighbors, which is what lets a group of 1s eliminate the variable that changes across it. The grid also wraps: left and right edges are adjacent, as are top and bottom.
Grouping rules
- Groups must be rectangles containing 1, 2, 4, 8, ... cells, always a power of two.
- Bigger groups eliminate more variables, so make groups as large as legal.
- Groups may overlap, and they may wrap around the edges.
- Cover every 1 with at least one group; ignore the 0s.
- Each group becomes one product term in the sum-of-products result.
Don't-care conditions
Some input combinations never occur or have no defined output. Marked with an X, these don't-cares can be treated as 1 or 0 freely, whichever makes groups larger and the expression simpler.
Reading a group
Within a group, keep the variables that stay constant and drop the ones that change. A group of two drops one variable, a group of four drops two, and so on. OR the resulting terms for the minimized function.
Limits
K-maps are ideal for three or four variables and workable at five or six with care. Beyond that, tabular methods such as Quine-McCluskey or automated logic synthesis take over.