Which Variables go on which side of a Karnaugh Map

Question

For a Karnaugh map of three or more variables deciding which side the variables go makes the solution easier to spot and simpler. But how do you know which side which variables go on.

eg. For variables x, y and z; You could have x and y as column headers and z as a row header or you could have y and z as column headers and x as a row header which would give two different tables

Answer 1

For maps with up to four variables, it is a matter of taste, which variable is put at which side. However, Mahoney maps as extension of Karnaugh maps for five and more variables do require a certain ordering along the side.

Expression for the following examples:

abcd!e + abc!de

Five-input Mahoney map:

Equivalent Karnaugh map:

            de                     de
      00  01  11  10         00  01  11  10
 abc +---+---+---+---+  abc +---+---+---+---+
 000 | 0 | 0 | 0 | 0 |  001 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+
 010 | 0 | 0 | 0 | 0 |  011 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+
 110 | 0 | 0 | 0 | 0 |  111 | 0 | 1 | 0 | 1 |
     +---+---+---+---+      +---+---+---+---+
 100 | 0 | 0 | 0 | 0 |  101 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+

It is always possible to swap variables as shown here:

           de                     de
      00  01  11  10         00  01  11  10
 abc +---+---+---+---+  abc +---+---+---+---+
 000 | 0 | 0 | 0 | 0 |  001 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+
 010 | 0 | 0 | 0 | 0 |  011 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+
 110 | 0 | 0 | 0 | 0 |  111 | 0 | 1 | 0 | 1 |
     +---+---+---+---+      +---+---+---+---+
 100 | 0 | 0 | 0 | 0 |  101 | 0 | 0 | 0 | 0 |
     +---+---+---+---+      +---+---+---+---+

Here you can find a nice online-tool to draw and simplify Karnaugh-Veitch/Mahoney maps.

Answer 2

The best approach to deciding which variables go to which side of the K-map is by first identifying the most significant variables and then putting that on the rows. For example;

For variables X, Y, and Z, if you regard X and the MSB and Z as the least, then you use X as the row header and YZ as the column header.