A2 · Practice questions · Karnaugh maps

Bigger groups, simpler terms.

Six original Cambridge-style questions on drawing and using Karnaugh maps for three and four inputs, with mark-scheme style solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

How to earn the marks

Map, group, read off.

Layout: label rows and columns 00, 01, 11, 10 and fill the 1s from the truth table.
Groups: 1, 2, 4 or 8 cells, rectangles only, as large and as few as possible, wrapping and overlapping allowed.
Terms: keep only the inputs that stay the same within each group.
Answer: OR the terms; one mark is often for each correct group or term.

Recall

[2 marks]

State two benefits of using a Karnaugh map to simplify a logic expression.

It is a systematic / visual method, so there is less chance of making an error than with Boolean algebra. ✓
It gives the simplest (minimal) sum-of-products expression, and it is quicker for three or four inputs. ✓

Apply

[3 marks]

A logic circuit has inputs A, B and C. The output X is 1 for the rows 000, 010, 100, 101 and 110 only. Draw the K-map and find the simplest expression for X.

K-map:

A \ BC 00 01 11 10

0 1 0 0 1

1 1 1 0 1

✓
Group of four: columns 00 and 10 wrap round (C = 0 throughout), giving C. ✓
Pair: row A = 1, columns 00 and 01 (A = 1, B = 0), giving A.B. X = C + A.B. ✓

A \ BC	00	01	11	10
0	1	0	0	1
1	1	1	0	1

Apply

[3 marks]

Find the simplest sum-of-products expression for this K-map.

AB \ CD	00	01	11	10
00	1	1	1	1
01	0	1	0	0
11	0	1	0	0
10	1	1	1	1

Rows AB = 00 and AB = 10 are neighbours (top and bottom wrap round) and are all 1s: a group of eight with B = 0, giving B. ✓
Column CD = 01 is all 1s: a group of four with C = 0 and D = 1, giving C.D. ✓
X = B + C.D ✓

Apply

[3 marks]

Find the simplest sum-of-products expression for this K-map.

AB \ CD	00	11	10
00	1	0	1
01	1	1	1
11	1	1	1
10	1	0	1

Columns CD = 00 and CD = 10 wrap round to make a group of eight: D. ✓
Cells 6, 7, 14, 15 make a group of four with B = 1 and C = 1: B.C. ✓
X = D + B.C ✓

Analysis

[4 marks]

A sprinkler X switches on when the water tank is not empty (B = 1) and at least one of these is true: the soil is dry (A = 1), the temperature is high (C = 1) or the timer is on (D = 1). Use a Karnaugh map to find the simplest sum-of-products expression for X.

K-map with 1s in cells 5, 6, 7, 12, 13, 14 and 15:

AB \ CD 00 01 11 10

00 0 0 0 0

01 0 1 1 1

11 1 1 1 1

10 0 0 0 0

✓
Row AB = 11 is a group of four: A.B. ✓
Middle two rows, columns 01 and 11: B.D; middle two rows, columns 11 and 10: B.C. ✓
X = A.B + B.C + B.D ✓

AB \ CD	00	01	11	10
00	0	0	0	0
01	0	1	1	1
11	1	1	1	1
10	0	0	0	0

Explain

[3 marks]

A student draws a three-input K-map with columns labelled 00, 01, 10, 11 and circles a group of three 1s. Explain the two errors and why each matters.

Columns must be in Gray code order 00, 01, 11, 10, so that neighbouring cells differ in only one input. ✓
With 01 next to 10, two inputs change between neighbours, so a group across them does not simplify correctly. ✓
Groups must contain 1, 2, 4 or 8 cells; a group of three does not correspond to a single product term (it must be split into overlapping pairs or enlarged). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.