K-map group size

Also called: group size, group size rule, power-of-two group size

A K-map group must contain 1, 2, 4, 8 or 16 cells. A group of 2ᵏ cells in an n-variable map gives a product term with n − k literals.

Every group must hold a power of two cells: 1, 2, 4, 8 or 16. No 3s, 5s or 6s.

Why powers of two? Each time a group doubles, it pairs every cell with a neighbor that differs in one more variable, and that variable cancels. A pair removes one variable, a group of 4 removes two, a group of 8 removes three. No product term covers exactly 3 cells, because a term either fixes a variable or doesn't, which always halves or keeps the count.

The size tells you the term length. In an n-variable map, a group of 2ᵏ cells gives n − k literals. For a 4-variable map:

1 cell → 4 literals
2 cells → 3 literals
4 cells → 2 literals
8 cells → 1 literal
16 cells → the constant 1

So bigger is better: each doubling removes a literal and makes the circuit cheaper. Always try to double a group before you accept it, but never by taking in a 0.

It also works in reverse. A term with L literals in an n-variable map covers 2ⁿ⁻ᴸ cells. Common names for these sizes are pairs, quads and octets.

A\BC	00	01	11	10
0	0m0	1m1	1m3	0m2
1	0m4	1m5	1m7	0m6

Worked examples

Example

From term to group size

In a 4-variable map, how many cells does cover, and which are they?

1.
It has 2 literals in a 4-variable map, so it covers 2⁴⁻² = 4 cells.
2.
A = 0 and D = 1, with B and C free: 0001, 0011, 0101, 0111.
3.
Those are m1, m3, m5 and m7, a 2×2 square in the top two rows. ✓

Example

From group to term length

The map above shows a group of 4 in a 3-variable map: the square m1, m3, m5, m7.

1.
4 = 2² cells, so k = 2 and the term has 3 − 2 = 1 literal.
2.
A changes and B changes; C stays 1. Term: C. One literal. ✓

Common mistakes

Circling 3 or 6 cells as one group. Split them into overlapping groups of 2 or 4.
Thinking a bigger group gives a longer term. It's the reverse: each doubling removes a literal.
Using n − k with k as the number of cells. k is the exponent: a group of 8 has k = 3.

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