# Karnaugh map with don't care: increasing the number of groups instead of simplifying

              AB
00  01  11  10
00 | x | 1 | 0 | 1 |
CD  01 | 0 | 1 | x | 0 |
11 | 1 | x | x | 0 |
10 | x | 0 | 0 | x |


The answer to the above Karnaugh map is $F(ABCD) = B D + \bar B \bar D + \bar A \bar C \bar D + \bar A C D$ according to my book and the K-map solvers online.

But what I don't get is that I can further reduce the terms in this answer: $F(ABCD) = \bar B \bar D + \bar A CD + \bar A B \bar C$ also covers all the min terms in lesser groups.

The answer in my book just has a bigger group covering additional don't cares. Since they are optional shouldn't my answer be correct?

To have optimal behavior with "don't care", you have to consider that an X can be either a 1 or a 0. That means that for each X, you have two versions of you map. So, if you have $N$ "don't cares", you will have to reduce the $2^N$ versions of the map and choose the smallest reduced operation from the results.