Boolean Algebra · step 11 of 13

Quine–McCluskey minimisation

Quine–McCluskey does what a Karnaugh map does, but as a table — so it scales past four variables and can be programmed. You group minterms by the number of 1s in their binary form, combine pairs differing in exactly one bit, repeat until nothing combines, and then select a minimal set of the resulting prime implicants using a covering chart.

Practise quine–mccluskey minimisation free → All Boolean Algebra subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Group minterms by 1-countWrite each minterm in binary and sort into groups by how many 1s it contains. Only adjacent groups can combine.
Combine pairs differing in one bitReplace the differing bit with a dash. Tick both parents — anything unticked at the end is a prime implicant.
Repeat on the new columnCombine again, matching dash positions. Stop when a pass produces nothing.
Build the covering chartRows are prime implicants, columns are original minterms. Pick essential prime implicants first, then cover the rest minimally.

Worked example

Find the prime implicants of f = Σm(0,1,2,3) over three variables x, y, z.

Binary: 0 = 000, 1 = 001, 2 = 010, 3 = 011. Group by 1-count: {000}, {001, 010}, {011}.
Combine 000 with 001 → 00–; 000 with 010 → 0–0.
Combine 001 with 011 → 0–1; 010 with 011 → 01–.
Second pass: 00– with 01– → 0––; 0–0 with 0–1 → 0––. All four first-level terms are ticked.

Answer. The single prime implicant is 0–– , i.e. x′ — so f = x′.

Where marks get dropped

These are the specific errors that cost credit on quine–mccluskey minimisation questions — QED's rubric penalises each of them separately.

Combining terms differing in more than one bit. Exactly one position may change, and dashes must align.
Forgetting to tick combined terms. Unticked entries at the end are the prime implicants, so bad bookkeeping loses them.
Stopping at the prime implicants. The covering chart is a separate step, and the minimal cover is what the question wants.

Practise this until it is automatic

Unlimited fresh questions

QED generates new quine–mccluskey minimisation problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Boolean Algebra mastery is tracked so you know when this is exam-ready.

Quine–McCluskey minimisation — frequently asked questions

Why use this over a Karnaugh map?

Because it is systematic and mechanisable. K-maps rely on visual adjacency, which collapses beyond four or five variables; Quine–McCluskey is just table manipulation.

Is it efficient?

Not asymptotically — the number of prime implicants can grow exponentially, and the covering step is NP-hard. Real tools use heuristics like Espresso.

What is a Petrick expansion?

An algebraic method for solving the covering chart exactly when no choice is forced: write a product of sums of implicants and multiply out to enumerate all minimal covers.

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make quine–mccluskey minimisation exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →