A simple well-known game, tic-tac-toe, is played on a three-by-three grid of squares by tw...

A simple well-known game, tic-tac-toe, is played on a three-by-three grid of squares by two players. The players alternate turns. Each player chooses a square and places a mark in a square. (One player uses X and the other O.) The first player with three marks in a row, in a column, or on a diagonal wins the game. A logic circuit is to be designed for an electronic tic-tac-toe that indicates the presence of a winning pattern. The circuit output W is a 1 if a winning pattern is present and a 0 if a winning pattern is not present. For each of the nine squares, there are two signals, Xi and Oi. Two copies of the circuit are used, one for Xs and one for Os. Hint: Form a condensed truth table for W(X1, X2,… X9).

(a) Design the X circuit for the following pattern of signals for the squares:

 

(b) Minimize the W output for the X circuit as much as possible, using Boolean algebra.