The Quine–McCluskey tabular method

0 · Recall Warm-up

Answer from memory first. Answers are in the Answer Vault at the bottom.

• What is an essential prime implicant? from Lesson 4.1.2
• Can a group in a K-map consist only of don't-cares? from Lesson 4.1.3
• Which rule lets two terms differing in one variable merge? from Lesson 3.1.2

1 · The Hook

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K-maps rely on your eyes, and eyes fail beyond 5 or 6 variables. In the 1950s Quine and McCluskey turned the same idea, "merge terms that differ in one bit", into a table a person (or program) can grind through. GATE names it in the syllabus as the tabular method and asks questions about its steps and its results.

2 · Predict First

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Can the minterms 0101 (5) and 0111 (7) be merged into one term? What about 0101 (5) and 0110 (6)?

Make a guess in your head. The answer opens Section 3.

3 · Core Concept

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Predict First — the answer

5 and 7 differ only in the third bit → merge to 01-1 (A'BD). 5 and 6 differ in two bits → cannot merge.

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In plain words

Write every minterm in binary. Repeatedly merge pairs that differ in exactly one bit, replacing that bit with a dash (-). Terms that can no longer be merged are the prime implicants. Then use a chart to choose the fewest prime implicants that cover all the minterms.

1. Step 1: group by number of 1s

Only terms in neighbouring groups (k ones and k + 1 ones) can differ in exactly one bit, so group them first.

2. Step 2: merge, column after column

• Compare each term with each term of the next group. If they differ in exactly one bit, write the merged term with a dash there, and tick both.
• In later columns, two terms may merge only if their dashes are in the same positions and they differ in one other bit.
• Repeat until nothing merges. Unticked terms are the prime implicants.
• Don't-cares are included in this step like minterms.

3. Step 3: prime implicant chart

Rows: the PIs. Columns: the minterms (not the don't-cares). Put an X where a PI covers a minterm.

• A column with a single X marks an essential PI.
• Remove the columns covered by EPIs; choose the fewest remaining PIs to cover the rest (Petrick's method when there is a real choice).

4. Worked example

F(A, B, C, D) = Σm(0, 1, 2, 5, 6, 7, 8, 9, 10, 14).

Prime implicants (unticked): -00- = B'C', -0-0 = B'D', --10 = CD', 0-01 = A'C'D, 01-1 = A'BD, 011- = A'BC. Chart: m₉ is covered only by B'C', and m₁₄ only by CD', so both are essential. They cover 0, 1, 2, 6, 8, 9, 10, 14. Left: 5 and 7, both covered by A'BD. F = B'C' + CD' + A'BD (3 terms, 7 literals).

4 · Visual

Merging rules

0101 (5)  and  0111 (7)   -> differ in 1 bit  -> 01-1   OK
0101 (5)  and  0110 (6)   -> differ in 2 bits -> no
0-01      and  0-11       -> dashes aligned, 1 other bit differs -> 0--1   OK
0-01      and  00-1       -> dashes in different places          -> no

5 · Worked Examples

Example 1 Easy

Can 1-10 and 1-11 be merged? Into what?

Solution

Dashes match, last bit differs → 1-1- (AC).

Example 2 Medium

After the merging steps, a term is left unticked. What is it?

Solution

A prime implicant (it cannot be enlarged).

Example 3 GATE

In the worked example, how many prime implicants are found, and how many are essential?

Solution

6 prime implicants, 2 essential (B'C' and CD').

6 · GATE Traps

⚠️

• Merge only terms differing in exactly one bit, with dashes in the same places.
• Don't-cares join the merging but are not columns of the chart.
• Every unticked term (in any column) is a prime implicant, not just those in the last column.
• The same merged term can arise from two different pairs; list it once.

7 · Memory Hook

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"Group by count of 1s, merge one-bit differences, dashes must line up; the unticked are prime; the chart picks the cover."

8 · Connect the Dots

Phase 4 is complete: we can find the cheapest two-level form of any function. Phase 5 builds real combinational circuits: adders, comparators, multiplexers and decoders.

9 · Self-Test

GATE format. Try every question before opening the vault.

Q1MCQ1 markMerging

Which pair can be merged in the Quine–McCluskey method?

• (A) 0-01 and 0-11
• (B) 1-0- and 1--0
• (C) 0-01 and 00-1
• (D) 0101 and 0110

Q2NAT1 markMerged term

The minterms 6 (0110) and 14 (1110) are merged. How many literals does the merged term have?

Numerical answer: ________

Q3NAT2 marksPrime implicants

F(A, B, C, D) = Σm(0, 1, 2, 5, 6, 7, 8, 9, 10, 14). How many prime implicants does the tabular method find?

Numerical answer: ________

Q4MCQ2 marksResult

The minimal SOP of F(A, B, C, D) = Σm(0, 1, 2, 5, 6, 7, 8, 9, 10, 14) is:

• (A) B'D' + A'C'D + A'BC
• (B) B'C' + B'D' + CD' + A'BD
• (C) B'C' + CD' + A'C'D + A'BC
• (D) B'C' + CD' + A'BD

Q5MCQ1 markChart

In the prime implicant chart, an essential prime implicant is found by looking for:

• (A) the longest term
• (B) a column with exactly one X
• (C) a row with no Xs
• (D) a row with the most Xs

10 · Rule Card

• Group minterms (and don't-cares) by number of 1s.
• Merge pairs differing in exactly one bit (dashes aligned); tick merged terms; repeat.
• Unticked terms = prime implicants.
• Chart (minterms only): single-X columns → EPIs; then cover the rest with the fewest PIs (Petrick if needed).

11 · Concept Ledger

Unlocked in this lesson. From here on, these terms may be used.

Quine–McCluskey methodgrouping by number of onesmerging terms with dashesticking merged termsprime implicants from the tableprime implicant chartessential PIs from single-X columnsPetrick's methoddon't-cares in the tabular method

12 · Answer Vault