Question

1. How many cells does a \(K{\bf{ - }}\)map in six variables have\(?\)
2. How many cells are adjacent to a given cell in a \(K{\bf{ - }}\)map in six variables\(?\)

Short answer

\((a)\)Six variables contain\(64\) cells

\((b)\)There are \(6\) adjacent cells

Step-by-step solution

Step 1:Definition

Product rule: If one event can occur in \(m\) ways and a second event can occur in \(n\) ways, then the number of ways that the two events can occur in sequence is then \(m \cdot n\).

Using product rule

A \(\)map for a function in six variables will contain \(2\) possible values for each variable \(x\) and \(\bar x\) for every variable \(x\).

Use the product rule:\(\underbrace {2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}_{6{\rm{ repetitions }}}{\bf{ = }}{{\bf{2}}^{\bf{6}}}{\bf{ = 6}}4\)

A \(K{\bf{ - }}\)map for a function in six variables then contains \(64\) cells.Since the \(K{\bf{ - }}\)map has \(6\) variables and since cells are adjacent if they differ in one variable thus a variable \(x\) is changed to \(\bar x\) or a variable \(\bar x\) is changed to

Therefore, one adjacent cell per variable and thus there are \(6\) adjacent cells.