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Pairs, quads and octets

Also called: K-map pair, quad, K-map quad, K-map octet, pair quad octet

Common names for K-map groups of 2, 4 and 8 cells. A pair removes one variable, a quad removes two and an octet removes three.

Many textbooks name K-map groups by size:

  • a pair is 2 adjacent cells and removes 1 variable
  • a quad is 4 cells, a row, column or 2×2 square, and removes 2 variables
  • an octet is 8 cells, two rows or two columns, and removes 3 variables

In an n-variable map, the term from each has n − 1, n − 2 or n − 3 literals. In a 4-variable map that means 3, 2 or 1 literal.

The names are handy when comparing options: "this 1 can join a pair or a quad" tells you at once that the quad gives the shorter term. The underlying rule is the k map group size rule: every doubling removes one more variable.

A quad or octet can be split across an edge by the wrap. The four corners of a 4-variable map are a quad, for example.

A single isolated cell is sometimes called a single or a monomial group; it removes nothing. A group of 16 covers the whole map and gives the constant 1.

AB\CD00011110
00
1m0
1m1
1m3
1m2
01
1m4
1m5
1m7
1m6
11
0m12
1m13
1m15
0m14
10
0m8
0m9
0m11
1m10

Worked example

Example

An octet, a quad and a pair

Simplify F = Σm(0–7, 10, 13, 15), shown above.

  1. 1.

    The top two rows are all 1s: an octet where A = 0 → . One literal.

  2. 2.

    m13 and m15 join m5 and m7 in a quad where B = 1 and D = 1 → . Two literals.

  3. 3.

    m10 (1010) has one neighboring 1, m2, across the top-bottom wrap. A pair → . Three literals.

  4. 4.

    F = .

Common mistakes

  • Calling any 4 cells a quad. They must form a rectangle, possibly wrapped.

  • Stopping at a pair when the same 1 could join a quad.

  • Mixing up which removes how many variables. A pair removes 1, a quad 2, an octet 3.

Practice Pairs, quads and octets

Interactive questions with instant feedback and a worked solution for every wrong answer.

Learn it step by step

Pairs, quads and octets is taught in Karnaugh Maps.