4-Variable Karnaugh Maps and the Four Corners

Sixteen minterms need a 4x4 grid: two variables assigned to the rows, 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 1

The 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. 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, everywhere else it's 0. Select all four corner cells and check the group.

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, and by six variables the wraparound relationships get hard to track by eye even with a lot of practice. That's 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.