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OCR H446 1.4.3 Three-variable Karnaugh maps

Part 6 of 11 · H446 1.4.3 · Boolean algebra

Eight input states, Gray-code column order and wraparound adjacency are what make three-variable maps harder than two, and OCR H446 1.4.3 expects all three. This worksheet decodes a cell before anything is placed in it, then builds towards overlapping and edge-wrapping groups.

Students will:

  • combine a column heading and a row heading to recover a complete three-variable state
  • explain why the headings run 00, 01, 11, 10 rather than in binary counting order
  • place an expression on a map term by term, treating absent variables correctly
  • form the largest legal groups, including a group that wraps across the edge
  • justify why a single 1 may belong to more than one group

Inside: 8 explanation cells, 5 multiple-choice questions, 3 fill-in-the-blanks cells and 1 written answer. 30 marks, about 45 to 55 minutes.

Series: H446 1.4.3 · Boolean algebra, part 6 of 11.

Shared by Coding PathwayVerified teacher

  • 17 cells
  • About 45 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.

Three-variable Karnaugh maps

A three-variable truth table has eight input states, so its Karnaugh map has eight cells. Two variables label the columns and the third labels the rows.

You should already know: that a cell is selected by its input-variable states and that a group removes changing variables.

Decode a cell before putting anything in it

With AB across the columns and C down the rows, combine the column header with the row header to recover the full state:

C / AB00011110
0ABC=000ABC=010ABC=110ABC=100
1ABC=001ABC=011ABC=111ABC=101

For example, column AB=10 and row C=1 is state ABC=101. If the output for truth-table row 101 is 1, place 1 there. Notice that the visual column order is not ordinary binary order.

Multiple choice1 mark

Where is truth-table state ABC=011 placed?

  • Acolumn AB=11, row C=0
  • Bcolumn AB=01, row C=0
  • Ccolumn AB=01, row C=1
  • Dcolumn AB=10, row C=1

Why the headings use Gray-code order

Use 00, 01, 11, 10, not 00, 01, 10, 11. Each move to a neighbouring column changes only one variable. The first and last columns are neighbours too: 00 and 10 differ only in A.

That one-change rule is what allows a group to remove a variable. If two variables changed at once, the group would not show that one input is irrelevant.

Multiple choice1 mark

Why is the column order 00, 01, 11, 10 used?

  • AIt places all 1s on the right
  • BIt is normal binary counting
  • CIt prevents groups from overlapping
  • DIt makes every adjacent pair differ in one variable

Worked placement: split at OR, then test each term

For (¬C ∧ B) ∨ (A ∧ B), treat the OR-separated product terms independently.

  1. For ¬C ∧ B, C must be 0 and B must be 1. A is absent, so its state does not matter: place 1 at ABC=010 and 110.
  2. For A ∧ B, A and B must both be 1. C is absent, so its state does not matter: place 1 at ABC=110 and 111.
  3. State 110 is covered twice but a cell still contains the single output 1.
  4. Any cell not made true by either term receives 0.

This ‘absent variable can be either state’ rule is central to placing expressions accurately.

Fill in the blanks8 marks
Complete the map for ¬C ∧ B ∨ A ∧ B. Entries are listed vertically for narrow screens; the coordinate model above remains the map layout. - C=0, AB=00: C0 AB00 - C=0, AB=01: C0 AB01 - C=0, AB=11: C0 AB11 - C=0, AB=10: C0 AB10 - C=1, AB=00: C1 AB00 - C=1, AB=01: C1 AB01 - C=1, AB=11: C1 AB11 - C=1, AB=10: C1 AB10
Written answer5 marks

On paper, draw the largest groups for the completed map. Derive the simplified term from each group and give the final expression.

One horizontal pair and one vertical pair are needed; they overlap at one cell.

Students type their answer here.

Read a different map

C / AB00011110
00011
10011

All four 1s form one rectangle. Across it, A stays 1; B and C both change.

Multiple choice1 mark

What is the simplest expression for this map?

  • AA
  • BB
  • CA ∧ B
  • DA ∨ C

Independent placement check

For A ∨ (¬B ∧ C), first fill every state with A=1. Then add states where B=0 and C=1; A is absent from the second term.

Fill in the blanks8 marks
Complete the map for A ∨ (¬B ∧ C). Use the same Gray-code coordinates as the model. - C=0, AB=00: entry 1 - C=0, AB=01: entry 2 - C=0, AB=11: entry 3 - C=0, AB=10: entry 4 - C=1, AB=00: entry 5 - C=1, AB=01: entry 6 - C=1, AB=11: entry 7 - C=1, AB=10: entry 8
Multiple choice1 mark

After optimal grouping, which expression is read from the independent map?

  • AA ∧ ¬B ∧ C
  • BA ∨ (¬B ∧ C)
  • CA ∧ C
  • D¬A ∨ B
Multiple choice1 mark

Why may one 1 appear in more than one Karnaugh group?

  • AEvery 1 must be counted twice
  • BGroups must cross diagonally
  • COverlap is allowed when it enables larger legal groups and a simpler expression
  • DIt changes the 1 into a don't-care value

Closed-book checkpoint

Retrieve the layout logic, not the completed example.

Fill in the blanks4 marks
A three-variable map has entry 1 cells. Its two-bit headings use entry 2 code order so neighbours differ in one bit. The first and last columns are also entry 3. Repeated placement leaves an occupied cell entry 4.

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.