Q.Using the four-variable K-map shown below, simplify the expression and show how encircling overlapping groups (allowing a 1 to be used in more than one group) gives a simpler result than not overlapping.

The map has a vertical octet of $D=1$ cells and a second quad of $B=1,C=1$ cells that share two 1s. Reusing (overlapping) those shared 1s lets the second group stay a full quad. …

[!TLDR]

Reusing a 1 in more than one group (overlapping) lets you build the largest groups, giving the minimal $Y = D + BC$ instead of the longer $Y = D + BCD$.

The map

Overlapping groups

An overlapping group is one that reuses a 1 already claimed by another group. The rule is to make each group as large as possible, so a 1 may belong to several groups.

Group 1 (octet): the whole $\overline{C}D$ and $CD$ columns — eight 1s across all four rows. Only $D$ stays constant ($D=1$); $A$, $B$ and $C$ all change, so this group reduces to $D$.

Group 2 (quad): the four 1s at rows $\overline{A}B$ and $AB$, columns $CD$ and $C\overline{D}$. Here $B=1$ and $C=1$ stay constant while $A$ and $D$ change, so this group reduces to $BC$. …