# Minimization and K-maps: GATE CS question with solution

From the GATE Daily set of 26 October 2026, question 4 of 10 (https://gatedaily.in/q/2026-10-26-kmap-corners-and-centre-minimal-sop/). Difficulty: easy.

### Q.4 · Digital Logic · Minimization and K-maps · MCQ, 2 marks (choose one option)

Consider the 4-variable Boolean function

$$F(w,x,y,z)=\Sigma m(0,2,5,7,8,10,13,15)$$

where $w$ is the most significant bit and $z$ the least significant bit, and there are no don't-care terms. Which one of the following is the **minimal** sum-of-products form of $F$?

(Here $'$ denotes complement.)

(A) $x'z' + xz$
(B) $x'z + xz'$
(C) $w'x'z' + wx'z' + xz$
(D) $x'z' + xyz$

**Answer:** (A) $x'z' + xz$

**Solution.** **Method: place the minterms on the K-map and take the largest groups.**

With rows $wx$ and columns $yz$ in Gray order ($00, 01, 11, 10$):

| $wx \backslash yz$ | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 00 | 1 (m0) | | | 1 (m2) |
| 01 | | 1 (m5) | 1 (m7) | |
| 11 | | 1 (m13) | 1 (m15) | |
| 10 | 1 (m8) | | | 1 (m10) |

- m0, m2, m8, m10 are the **four corners**. The map wraps left-right and top-bottom, so they form one group of four where $x = 0$ and $z = 0$: the term $x'z'$.
- m5, m7, m13, m15 form the central $2 \times 2$ block where $x = 1$ and $z = 1$: the term $xz$.

Both groups are essential prime implicants and together cover all eight minterms, so $F = x'z' + xz$ (that is, $F = 1$ exactly when $x = z$).

**Trap.** Not seeing that the corners are adjacent leads to two separate pairs ($w'x'z' + wx'z'$), which is correct as a function but has more literals.

**What was tested:** wrap-around adjacency on a 4-variable K-map.

**Remember:** The four corners of a 4-variable K-map form a valid group of four through horizontal and vertical wrap-around (diagonally opposite corners are not adjacent to each other, but the group is still legal). Check wrap-around groups before settling for pairs.

**Why each option:**
- (A) The four corners give $x'z'$ and the central block of four gives $xz$; both are essential and cover everything.
- (B) This is the complement pattern (x XOR z); it comes from reading the 1-cells as 0-cells.
- (C) Equals F, but splits the four corners into two pairs; it misses the wrap-around and is not minimal.
- (D) Covers only minterms 7 and 15 of the centre block, so it misses 5 and 13 and is not F.