5.4 4-Variable K-map: 16 Cells, Wrap-Around Groups, and Essential Groups
Build a 16-cell, 4-variable K-map with Gray-coded rows and columns, identify every wrap-around adjacency (including the four-corner group), and select a minimal cover using essential groups when multiple grouping choices exist.
The 4-variable K-map is the largest map most engineers ever draw by hand, and it is the map where every rule from the smaller maps must work together at once: Gray-code rows and columns, horizontal and vertical wrap-around, and — uniquely at this size — a wrap-around that connects all four corners of the grid simultaneously. It handles any Boolean function of four inputs, which covers a large share of practical small combinational logic (such as 4-bit comparators, BCD-related logic, and small control decoders), making it the workhorse map of introductory digital design.
16-cell layout: two Gray-coded axes
A 4-variable map splits the four inputs into two pairs: one pair (commonly A, B) labels the four rows in Gray-code order (00, 01, 11, 10), and the other pair (commonly C, D) labels the four columns, also in Gray-code order (00, 01, 11, 10). Every cell is addressed by its row pair and column pair together, giving all 16 minterms (m0 through m15) a unique position. Because both axes are Gray-coded, moving one step in any direction — up, down, left, or right, including wrapping past an edge — changes exactly one of the four variables.
This double Gray-coding is what makes the 4-variable map tractable: without it, a 16-cell grid would have far too many false 'near misses' where visually close cells differ in two or more variables and cannot be grouped.
Wrap-around groups, including the four corners
Just as in the smaller maps, the leftmost and rightmost columns are adjacent (Gray-code columns 00 and 10 are one-bit neighbors), and the top and bottom rows are adjacent for the same reason. In the 4-variable map this produces a distinctive extra case: the four corner cells (m0, m2, m8, m10) are all mutually reachable through wrap-around — the top-left cell is adjacent to the top-right cell (row wrap), and separately adjacent to the bottom-left cell (column wrap), and the four corners together can form a single valid quad group.
This corner-to-corner group is easy to miss because the four cells are maximally far apart visually, in the four corners of the drawn grid, yet they are a perfectly legal power-of-two (4-cell) rectangle once both wrap-arounds are accounted for. Two of the four input variables (B and D, in the common A,B row / C,D column convention) stay constant at 0 across all four corner cells, while A and C both vary, so the corner group eliminates A and C and leaves the two-literal term B'D'.
Multiple grouping choices and picking essential groups
A 4-variable map frequently offers more than one way to group the same set of 1s into valid prime implicants — for example, a single 1 might be coverable by two different, equally large groups. When this happens, the systematic procedure is: first identify every prime implicant on the map (every group that cannot be enlarged), then identify which prime implicants are essential (the only group covering some particular minterm) and include all of them unconditionally, and only after that choose among the remaining, non-essential prime implicants the smallest additional set needed to cover any minterms not yet covered.
Essential groups are never optional and never subject to a 'better-looking' alternative — if a minterm has only one possible covering group, that group must appear in the final answer regardless of whether a different grouping choice would look tidier elsewhere on the map. Choosing between genuinely tied, non-essential options is where an engineer has freedom, typically resolved by whichever choice results in fewer total literals or fewer total product terms.
4-variable K-map layout (AB on rows, CD on columns, both Gray-coded)
Always draw both axes in Gray-code order 00, 01, 11, 10 — never plain binary order 00, 01, 10, 11. The four-corner group is the signature 'gotcha' unique to the 4-variable map; smaller maps don't have enough cells to produce it.
CD=00 CD=01 CD=11 CD=10
AB=00 [ m0 ] [ m1 ] [ m3 ] [ m2 ]
AB=01 [ m4 ] [ m5 ] [ m7 ] [ m6 ]
AB=11 [ m12 ] [ m13 ] [ m15 ] [ m14 ]
AB=10 [ m8 ] [ m9 ] [ m11 ] [ m10 ]
Row order and column order both follow Gray code: 00, 01, 11, 10
Row 00 is adjacent to Row 10 (top/bottom wrap); Column 00 is adjacent to Column 10 (left/right wrap)
=> the four corners m0, m2, m8, m10 are mutually adjacent through wrap-around.Code Examples
This is exactly the spec's tricky problem in action: a corner-to-corner group is valid because K-map edges wrap. The four corners are reached by combining the row wrap (top row adjacent to bottom row) and the column wrap (leftmost column adjacent to rightmost column) at the same time, and the resulting quad eliminates A and C while keeping B'D'. Expected result: F(A,B,C,D) = B'D'
Corner cells m0, m2, m8, m10 form a valid wrap-around quad, simplifying to B'D'.
Here the four 1s form a single unambiguous quad with no competing grouping choice, illustrating the 'easy' case. When maps are denser, the same minterm can sometimes be covered by two different, equally valid quads — in that situation you check whether either quad is essential (the only one covering some other minterm); if neither is essential on its own, either may be chosen, guided by which choice minimizes the total number of groups needed for the whole map.
Systematically identify all prime implicants first, then pick essential ones before resolving any remaining ties.
CD=00 CD=01 CD=11 CD=10
AB=00 [ 1 ] [ 0 ] [ 0 ] [ 1 ]
AB=01 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
AB=11 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
AB=10 [ 1 ] [ 0 ] [ 0 ] [ 1 ]
Corners m0 (AB=00,CD=00), m2 (AB=00,CD=10), m8 (AB=10,CD=00), m10 (AB=10,CD=10) are all 1.
Across these 4 cells: A varies (0/1) -> eliminated. B stays 0 throughout -> kept as B'.
C varies (0/1) -> eliminated. D stays 0 throughout -> kept as D'.
Simplified SOP: F(A,B,C,D) = B'D'CD=00 CD=01 CD=11 CD=10
AB=00 [ 0 ] [ 1 ] [ 1 ] [ 0 ]
AB=01 [ 0 ] [ 1 ] [ 1 ] [ 0 ]
AB=11 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
AB=10 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
m1, m3, m5, m7 are all 1 and form one clean quad (columns CD=01 and CD=11, rows AB=00 and AB=01):
A stays 0, D stays 1 constant across the quad; B and C vary and are eliminated.
Simplified SOP: F(A,B,C,D) = A'D4-variable K-map with the corner group highlighted
The four corners of a 4-variable K-map form a legal quad because both the row axis and the column axis wrap around.
CD=00 CD=01 CD=11 CD=10
AB=00 [ (1) ] [ 0 ] [ 0 ] [ (1) ]
AB=01 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
AB=11 [ 0 ] [ 0 ] [ 0 ] [ 0 ]
AB=10 [ (1) ] [ 0 ] [ 0 ] [ (1) ]
(1) marks the four corner cells, mutually adjacent via top/bottom and left/right wrap-around.
This is the largest, and most visually surprising, wrap-around group possible on a 4-variable map.Important Rules
Both the row-pair and column-pair headings must be in Gray-code order: 00, 01, 11, 10.
Top row and bottom row are adjacent (wrap); leftmost and rightmost columns are adjacent (wrap).
The four corner cells (m0, m2, m8, m10 in the standard layout) are mutually adjacent and can form one valid quad.
Always find all prime implicants first, mark essential ones as mandatory, then resolve remaining minterms with the fewest additional non-essential groups.
Valid group sizes remain powers of two: 1, 2, 4, 8, 16.
Where 4-input logic is used in practice
Four-variable K-maps map directly onto real 4-bit digital building blocks: a 4-bit magnitude comparator, BCD-to-something decoders, and small 4-input control-logic blocks in a CPU's control unit are all naturally described by exactly four Boolean inputs. Being able to hand-minimize a 4-variable function is also the standard technique taught for deriving datasheet-published logic equations for small MSI (medium-scale integration) chips, and it remains the largest map size most textbooks and interviewers expect an engineer to minimize entirely by hand before reaching for computer-aided tools.
Ignoring the four corner cells as 'too far apart to matter' and grouping them individually as four separate single-cell groups.
Check whether the four corners are all 1; if so, group them together as a single valid quad using both wrap-arounds simultaneously.
The spec's tricky problem is explicit: a corner-to-corner group can be valid because K-map edges wrap. Missing it produces an unnecessarily large (non-minimal) expression with four single-literal-heavy terms instead of one two-literal term.
Picking a grouping for a minterm just because it looks larger or more convenient, without first checking whether it is essential.
Always identify essential prime implicants first (groups that are the only coverage for some minterm) and lock those in before choosing among any remaining, genuinely optional groupings.
Skipping the essential-prime-implicant check can lead to a valid-looking but non-minimal cover, or in rarer cases a cover that misses a minterm entirely if a supposedly 'obvious' larger group was chosen over the one truly essential group.
[tricky] On a 4-variable K-map, can the four corner cells (m0, m2, m8, m10 in the standard AB-row / CD-column layout) ever form a single valid group? A. Yes — because both the row axis and the column axis wrap around, the four corners are mutually adjacent and can form a legal quad B. No — corner cells are always diagonal to each other and therefore never adjacent C. Only if the function has fewer than 4 variables D. Only in a POS map, never in an SOP map
A. Yes — because both the row axis and the column axis wrap around, the four corners are mutually adjacent and can form a legal quad — The spec's tricky problem states this directly: a corner-to-corner group can be valid because K-map edges wrap. Both the top/bottom row wrap and the left/right column wrap apply simultaneously to the four corners, making them a legitimate 4-cell (quad) group.
When a 4-variable K-map offers multiple possible groupings for the same set of 1s, what is the correct first step? A. Identify all prime implicants, then include every essential prime implicant (the sole coverage for some minterm) before resolving any remaining, genuinely optional choices B. Always choose the group that appears first when scanning left to right, top to bottom C. Randomly choose any valid grouping, since all minimal covers are equally correct D. Avoid quads and octets entirely and only use pairs, to keep the derivation simple
A. Identify all prime implicants, then include every essential prime implicant (the sole coverage for some minterm) before resolving any remaining, genuinely optional choices — The systematic K-map procedure requires locking in every essential prime implicant first, because those are mandatory (no alternative covers those minterms). Only after that should any remaining, non-essential ties be resolved, typically by minimizing total literal or term count.
Minimize F(A,B,C,D) = Σm(0,2,8,10,5,7,13,15) using a 4-variable K-map. Identify both wrap-around groups present.
m0, m2, m8, m10 form the corner quad B'D' (as in the worked example above). m5, m7, m13, m15 occupy CD=01/CD=11 across AB=01 and AB=11 — check adjacency: B and D stay 1 constant across all four, A and C vary and are eliminated, giving BD. Full simplified SOP: F(A,B,C,D) = B'D' + BD.
Place all eight 1s on the map. Check the four corners as one group, and separately check whether m5, m7, m13, m15 form a second wrap-around (or non-wrap) quad.