Karnaugh Map Solver

That is reflected Gray code, G(i) = i ⊕ (i ≫ 1), and it buys exactly one property: adjacent cells differ in one variable. That is precisely the condition under which A·B + A·B′ = A applies, so a group of two is one use of the adjacency law, a group of four is two, and a group of eight is three. Label the axes in plain binary order and the groups stop being rectangles. The Gray code converter shows the reflection that generates the order.

Two to six variables, each with the function that makes that size worth understanding. These are anchors on this page rather than separate pages, because one concept fragmented across five URLs splits its own authority and teaches the reader nothing extra.

Above four variables the map becomes 2 or 4 planes of a four-variable map, never one wide strip. With three bits on one axis, 6 of the 27 subcubes are not contiguous runs and 4 contiguous runs are not subcubes — so a strip would both reject legal groups and accept illegal ones. Cells in the same position on adjacent planes are adjacent, which is what makes the eight-cell group in the five-variable example above a single term.

A don't-care is an input combination that cannot occur, or one whose output nobody cares about. Read each as 1 exactly when doing so enlarges a group you already need, and never make a group that contains only don't-cares: it is a legal implicant that adds a term while covering nothing the function requires. The engine enforces both halves of that — it flags a don't-care-only group, and it drops prime implicants that cover no ON-set minterm before the chart is built, which is a real failure mode of the naive algorithm.

Every result restates which don't-cares were read as 1 and which were left as 0, because two covers of identical cost can disagree about a don't-care and printing that difference is what makes them comprehensible.

• Prime — the group cannot be doubled.
• Essential — it is the only group that can cover some 1. That 1 is its distinguishing cell, and the map marks it.
• Irredundant — no group in your cover can be deleted.
• Minimum — no cover costs less.

An irredundant cover can still have more terms than the minimum. Conflating the two is a mistake several published tools make, and it is why the verdict here distinguishes “complete and irredundant, but not minimum” from “correct”.

The map opens on Σm(0, 1, 2, 5, 8, 9, 10) over A B C D. It has 3 prime implicants, 3 of them essential, and exactly one minimum cover: minimal SOP B′ · C′ + B′ · D′ + A′ · C′ · D, minimal POS (C′ + D′) · (B′ + D) · (A′ + B′). Those numbers come from the engine at build time, not from a template.

Other functions are not so tidy. Σm(0, 1, 2, 5, 6, 7) over three variables has six prime implicants, noneessential, and two equally minimal covers — the chart is cyclic, which is exactly the case Petrick's method exists for. When that happens this page shows every cover and says so in the verdict, because a student whose grouping differs from the tool's but costs the same is right and should be told so.

A Karnaugh map is drawn for 2 to 6 variables. Above six, every axis would need three bits, and on a three-bit Gray axis a rectangle and a subcube stop being the same thing — so the map would reject legal groups and accept illegal ones. Use Quine–McCluskey instead.

Within those two to six variables the minimisation is exact, not heuristic. Enumeration of equally minimal covers stops at 200 displayed and 200 counted, and the badge says so when it stops rather than printing a number the engine did not prove. Worst-case prime-implicant count grows as Θ(3ⁿ/√n), which is why the limits exist and why they are printed rather than hidden.