Simplifying Karnaugh Maps by Making Groups of a Non-Power-of-Two Number of Elements

TL;DR

This paper introduces a method to simplify Karnaugh maps by forming groups of non-power-of-two elements, such as three or $2^n-1$, resulting in simpler logic functions than traditional sum of products.

Contribution

It demonstrates that grouping non-power-of-two elements in Karnaugh maps can produce simpler logic functions, extending the traditional grouping rules.

Findings

Groups of three elements can simplify logic functions.

Extending to groups of $2^n-1$ elements further reduces complexity.

The approach offers an alternative to standard sum of products simplification.

Abstract

When we study the Karnaugh map in the switching theory course, we learn that the ones in the map must be combined in groups of $a \times b$ elements, being $a$ and $b$ powers of two. The result is the logic function described as a sum of products. This paper shows that we can also make groups where $a$ and/or $b$ are equal to three. This does not result in a sum of products, but in a logic function that is simpler than the sum of products in terms of logic gates. This idea is extended later in the paper to groups of $2^n-1$ elements.