Community resourceWorksheet
OCR H446 1.4.3 Four-variable Karnaugh maps
Part 7 of 11 · H446 1.4.3 · Boolean algebra
With 16 input states, grouping has to become a reliable routine rather than an intuition. Pitched at OCR H446 1.4.3, this worksheet sets out a step-by-step grouping procedure, works a wraparound pair on a full four by four map, and insists on the visible annotation that examination questions ask for.
Students will:
- transfer a truth-table row to the correct cell of a four-variable map
- work out how many cells a term makes true when some variables are absent
- follow a fixed routine that covers every 1 with the fewest, largest groups
- reject a proposed group that contains a 0 or breaks the power-of-two rule
- annotate a map with its groups and write the simplified expression
Inside: 10 explanation cells, 6 multiple-choice questions, 2 fill-in-the-blanks cells and 2 written answers. 26 marks, about 50 to 60 minutes.
Series: H446 1.4.3 · Boolean algebra, part 7 of 11.
Shared by Coding PathwayVerified teacher
- 20 cells
- About 60 minutes
- CC BY-SA 4.0
- Shared 31 Aug 2026
- Updated 15 Sept 2026
Preview
The whole resource, exactly as a class sees it. Answers and marking are held back.
Four-variable Karnaugh maps
A four-variable truth table has 16 input combinations, so its Karnaugh map is a 4 × 4 grid. OCR questions commonly ask you to transfer outputs, annotate groups and write the simplified expression.
You should already know: the heading order 00, 01, 11, 10; term-by-term placement; and how a group removes changing variables.
Worked model: use the row and column headings
Place AB across the columns and CD down the rows. Both sets of headings use 00, 01, 11, 10.
For F = A ∧ ¬D, A must be 1 and D must be 0. B and C may take either value:
| CD ↓ / AB → | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 00 | 0 | 0 | 1 | 1 |
| 01 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 1 | 1 |
The four output 1s form a top-to-bottom wrap-around group. B and C change inside the group; A remains 1 and D remains 0, giving A ∧ ¬D.
Use the headings to place each output. You do not need to write a four-bit input label inside every box.
A truth-table row has A=1, B=1, C=1 and D=0. Which box receives its output?
- Acolumn AB=10, row CD=11
- Bcolumn AB=11, row CD=01
- Ccolumn AB=01, row CD=11
- Dcolumn AB=11, row CD=10
Place an expression term by term
For (¬C ∧ D) ∨ (A ∧ B):
- ¬C ∧ D fixes row CD=01. A and B are absent, so fill all four boxes across that row with 1.
- A ∧ B fixes column AB=11. C and D are absent, so fill all four boxes down that column with 1.
- The intersecting box still contains one output 1. Every other box contains 0.
A product term containing four variables identifies one box; three variables identify two boxes; two variables identify four boxes; one variable identifies eight boxes. Each missing variable may be 0 or 1.
In a four-variable map, how many boxes are made true by the term A ∧ ¬D?
- A4
- B2
- C8
- D16
Worked model: a real pair across the edge
The two highlighted cells share B=0, C=0 and D=1. Only A changes, so they are logically adjacent even though the grid is drawn with a gap between them. Their group represents ¬B ∧ ¬C ∧ D.
The same rule applies vertically: top meets bottom because CD=00 and CD=10 differ only in C. The four corners can therefore form one legal group of four when all four contain 1.
A reliable grouping routine
A 1 is a true output and a 0 is a false output.
- Cover every 1 at least once.
- Use rectangles containing 1, 2, 4, 8 or 16 cells.
- Include only 1s.
- Make each group as large as possible.
- Use as few groups as possible, allowing overlap when helpful.
- For each group, keep only variables that stay constant.
Quick practice: read whole wrap-around groups
Map X has 1s in every cell of the first and last columns and 0s elsewhere. Across AB=00 and AB=10, A changes while B stays 0.
Map Y has 1s in every cell of the top and bottom rows and 0s elsewhere. Across CD=00 and CD=10, C changes while D stays 0.
What is the simplest expression for Map X?
- A¬A
- B¬B
- C¬C
- D¬D
What is the simplest expression for Map Y?
- AC
- B¬C
- C¬D
- DD
| CD ↓ / AB → | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 00 | row 1 column 1 | row 1 column 2 | row 1 column 3 | row 1 column 4 |
| 01 | row 2 column 1 | row 2 column 2 | row 2 column 3 | row 2 column 4 |
| 11 | row 3 column 1 | row 3 column 2 | row 3 column 3 | row 3 column 4 |
| 10 | row 4 column 1 | row 4 column 2 | row 4 column 3 | row 4 column 4 |
Independent paper task: annotate the completed map with the two largest groups. Write the simplified term for each group and the final expression.
One group joins the top and bottom rows; the other joins the left and right columns.
Students type their answer here.
Expression-to-map examination practice
On paper, draw a blank four-variable map and place (¬A ∧ B ∧ C) ∨ (B ∧ ¬D). Use AB across the columns and CD down the rows. Form the largest groups before selecting your answer.
Which simplified expression should your groups produce?
- A¬A ∧ B ∧ C
- BB ∧ ¬D
- C¬A ∨ B ∨ C ∨ ¬D
- D(¬A ∧ B ∧ C) ∨ (B ∧ ¬D)
A proposed group contains three 1s and one 0. Why must it be rejected?
- AA group may contain only 1s
- BGroups cannot contain four cells
- CGroups must use ordinary binary order
- DEvery group must wrap around
Explain why the first and last columns of a four-variable Karnaugh map are adjacent.
Compare the headings 00 and 10.
Students type their answer here.
Examination lens
OCR has awarded separate marks for correct groups, terms and the OR joining those terms. Annotate the map visibly. A correct final expression without shown grouping may lose method marks when the question asks for annotation.
Closed-book checkpoint
Complete these error-prevention rules from memory.
Review your work
Check that every 1 is covered, no group contains a 0 and no larger legal group has been missed.