4-Variable Karnaugh Maps and the Four Corners
A full 4x4 grid, Gray code running along both axes at once, and the classic four-corner group explained rather than just named.
Sixteen minterms need a 4x4 grid, two variables assigned to the rows and two to the columns, both axes labeled in the same 00, 01, 11, 10 Gray code order used for the 3-variable map's columns. Nothing new is happening mathematically here. There are just two Gray-coded axes running at once instead of one.
Group Sizes Get Bigger
With sixteen cells, legal group sizes are 1, 2, 4, 8, or the entire map as a single group of 16 if the function is a constant 1.
Group of 1 -> 0 variables cancel -> full 4-literal term
Group of 2 -> 1 variable cancels -> 3-literal term
Group of 4 -> 2 variables cancel -> 2-literal term
Group of 8 -> 3 variables cancel -> 1-literal term
Group of 16 -> 4 variables cancel -> constant 1The Four Corners
Because both axes wrap around independently, the four corner cells of a 4-variable map are mutually adjacent even though none of them touch on paper. The top-left cell is adjacent to the top-right by row-wraparound, adjacent to the bottom-left by column-wraparound, and reaches the bottom-right by combining both wraps at once, so all four corners are legally a single group of four whenever they all happen to be 1.
Try it below. F(A, B, C, D) is 1 whenever B = 0 and D = 0, and everywhere else it is 0. Select all four corner cells and check the group.
No groups yet.
4 cells to cover. Bigger groups mean fewer literals, so always take the largest legal group you can.
Every corner shares B = 0 and D = 0 while A and C vary across all four combinations between them, so those two variables cancel and only B'D' survives. Four cells, two variables gone, exactly the same power-of-two relationship as every other group size. It just happens to be scattered across all four corners of the printed grid instead of sitting in one visual block.
Corners: {A'B'C'D', AB'C'D', A'B'CD', AB'CD'}
All four share B = 0 and D = 0. A and C take every combination between them.
=> only B'D' survives -> F = B'D'Beyond Four Variables
A 5-variable map needs two 4x4 grids side by side. By six variables the wraparound relationships get hard to track by eye even with practice, which is the practical ceiling for doing Karnaugh maps by hand. Past it, tabular methods like Quine-McCluskey, or a computer searching the same adjacency structure, take over. The underlying idea never changes: find the largest legal power-of-two group covering each 1, and the smallest Sum-of-Products falls out.
A second example: overlapping groups
Six 1s. The largest groups available share cells, and that is fine, since covering a 1 twice costs nothing.
F = A'B + BD
6 ones, covered by 2 groups. This grouping is the only minimal one.
The answer is A'B + BD. Both groups are blocks of four, and they intersect.
People often try to partition the 1s into non-overlapping blocks, as though the map were a jigsaw. It is not. The only requirements are that every 1 is inside some group, no group touches a 0, and no group is redundant.