Cube notation

Cube notation writes a product term as a pattern of n symbols, one per variable in order:

• 1 for a plain variable,
• 0 for a barred variable,
• - for a variable that does not appear.

With inputs A, B, C: AC is 1-0, B is -1-, and the minterm ABC is 011.

Each dash can be 0 or 1, so a pattern with k dashes stands for 2ᵏ rows. 1-0 covers 100 and 110, rows 4 and 6. The name comes from geometry: the rows of an n-variable table are the corners of an n-dimensional cube, and a term with k dashes is a k-dimensional face of it.

Why it helps:

• Expanding a term to minterms is just filling in the dashes; see canonical expansion.
• Combining two patterns is easy to check: same dashes, one other bit different. The differing bit becomes a new dash. See adjacent minterms.
• Counting overlaps between terms becomes listing rows and taking the union.

For sum terms, the same pattern describes the rows where the sum is 0, read with maxterm polarity.