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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 3 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 states, 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: Gray-code order, term-by-term placement and how a group removes changing variables.

Worked model: every cell is one complete ABCD state

Every Karnaugh-map cell is one complete input statecolumn gives A and B00011110row gives C and D000111100000010011001000000101011101100100110111111110110010011011101010AB = 10 and CD = 01, so the complete state is ABCD = 1001The number written in the cell is the output for that state, not the state itself.

The column gives the states of A and B. The row gives the states of C and D. Join them in the variable order ABCD to identify the truth-table row.

At column AB=10 and row CD=01, A=1, B=0, C=0 and D=1. This is state 1001. The output for state 1001 (0 or 1) is what belongs inside the map cell.

Keep these two ideas separate: the headers identify the input state; the cell entry records the output.

Multiple choice1 mark

Which cell receives the output from truth-table row ABCD=1110?

  • 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 CD=01. A and B are absent, so fill all four cells across that row with 1.
  • A ∧ B fixes AB=11. C and D are absent, so fill all four cells down that column with 1.
  • The intersecting cell is still simply 1. Every other cell is 0.

A term containing four variables identifies one cell; three variables identify two cells; two variables identify four cells; one variable identifies eight cells. Each missing variable is free to be 0 or 1.

Multiple choice1 mark

In a four-variable map, how many cells are made true by the term A ∧ ¬D?

  • A4
  • B2
  • C8
  • D16

Worked model: a real pair across the edge

A concrete wrap-around pairAB00011110CD00011110state 00011state 10011B=0, C=0 and D=1 stay the same; only A changes.Therefore these two 1s form a legal edge group and give ¬B ∧ ¬C ∧ D.

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.

  1. Cover every 1 at least once.
  2. Use rectangles containing 1, 2, 4, 8 or 16 cells.
  3. Include only 1s.
  4. Make each group as large as possible.
  5. Use as few groups as possible, allowing overlap when helpful.
  6. 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.

Multiple choice1 mark

What is the simplest expression for Map X?

  • A¬A
  • B¬B
  • C¬C
  • D¬D
Multiple choice1 mark

What is the simplest expression for Map Y?

  • AC
  • B¬C
  • C¬D
  • DD
Fill in the blanks8 marks
The output is 1 for ordinary binary input rows ABCD = 0001, 0011, 1000, 1001, 1010, 1011, 1100 and 1110. Complete the Karnaugh map. Enter only 0 or 1. Coordinates are listed vertically in Gray-code row and column order for narrow screens. - CD=00, AB=00: row 1 column 1 - CD=00, AB=01: row 1 column 2 - CD=00, AB=11: row 1 column 3 - CD=00, AB=10: row 1 column 4 - CD=01, AB=00: row 2 column 1 - CD=01, AB=01: row 2 column 2 - CD=01, AB=11: row 2 column 3 - CD=01, AB=10: row 2 column 4 - CD=11, AB=00: row 3 column 1 - CD=11, AB=01: row 3 column 2 - CD=11, AB=11: row 3 column 3 - CD=11, AB=10: row 3 column 4 - CD=10, AB=00: row 4 column 1 - CD=10, AB=01: row 4 column 2 - CD=10, AB=11: row 4 column 3 - CD=10, AB=10: row 4 column 4
Written answer6 marks

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.

Multiple choice1 mark

Which simplified expression should your groups produce?

  • A¬A ∧ B ∧ C
  • BB ∧ ¬D
  • C¬A ∨ B ∨ C ∨ ¬D
  • D(¬A ∧ B ∧ C) ∨ (B ∧ ¬D)
Multiple choice1 mark

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
Written answer2 marks

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.

Fill in the blanks4 marks
A four-variable map contains entry 1 cells. A legal group is a entry 2 whose size is a power of two. Edge wrapping works because the relevant headings differ in only entry 3 bit. Groups must never include a entry 4 output.

Review your responses

Check every response against its command word and the supplied constraints. Strengthen unsupported answers with accurate method, mechanism, state or contextual consequence before submitting.