Electronic – Karnaugh map subcube with no unchanging variables

boolean-algebrakarnaugh map

I am tasked with using a 5-variable k-map to minimize a SOP expression. The minterms are 3,7,12,14,15,19,23,27,28,29,31, and the don't cares are 4,5,6,13,30. (Where, in the class I am taking, you start counting from zero)

There is one subcube that can be drawn that includes positions 3,4,5,6 and 27,28,29,30. This subcube has no unchanging variables for it… so what would its expression be?

This same problem can also appear in a 4 variable kmap if positions 5,6, and 9,10 are part of a subcube. What would you write for this?

Please note: this is a homework problem, so please do not solve the entire system for me. I am only asking because we did not learn in class what to do in this case, and I cannot find anything helpful online.

Best Answer

Lets say your five variables are A, B, C, D, E. To "draw" a 5 variable K-map, draw two K-Maps and imagine them stacked on top of one-another.

E=0

                !C          C
            ----------  ---------
     AB\CD  00    01    11   10
!A { 00     0000  0001  0011 0010  > !B
!A { 01     0100  0101  0111 0110  } B
A  < 11     1100  1101  1111 1110  } B
A  < 10     1000  1001  1011 1010  > !B
            vvvv  ---------- vvvv         
             !D       D       !D      


E=1
                !C          C
            ----------  ---------
     AB\CD  00    01    11   10
!A { 00     0000  0001  0011 0010  > !B
!A { 01     0100  0101  0111 0110  } B
A  < 11     1100  1101  1111 1110  } B
A  < 10     1000  1001  1011 1010  > !B
            vvvv  ---------- vvvv         
             !D       D       !D

You see how every grid location (which maps to a min-term) differs from all its neighbors (including wrap-around edge-cases) by exactly one bit? That's more-or-less the whole point of a K-Map representation. Now in a 4-variable K-Map you can "circle" contiguous rectilinear shapes where the output is a "1" so long as their "area" is a power of two (i.e. 1, 2, 4, 8, or 16 terms) in order to achieve a "cover" of the K-map. I believe there are 2^16 possible covers of a 4-variable K-map (i.e. 2^(2^4)).

The only difference in the 5-variable case is "adjacent" also include analogous positions in the two 4-variable maps (sub-spaces), and there are more ways to ways to "circle" a number of cells that is a power of two; that is to say there are "three-dimensional" covers possible.

Hopefully it's clear that "the thing that is constant in a straight line" is what that line identifies, and that the intersections of "the things that are constant" is what makes a particular location a minterm. If you "circle" the upper-left four grid locations in both E=0 and E=1 maps, then E is a don't-care. If you circle the quadrant in just the E=0 map, then the term you are circling must have E in the intersection.

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