Math Problem Solver

Each of the seven boxes in the figure is to be filled with one of the letters A, B, or C.

A box is gregarious if the letter it contains, together with the letters in the neighboring

boxes, include all three of the letters A, B, and C. For example, in the example on the

right, the box in the middle containing an A is gregarious, since it contains an A and

its neighbors contain at least one each of B and C, but the box on the lower right corner

is not gregarious, since it neither contains nor neighbors an A. How many ways are

there to fill the boxes so that all seven boxes are gregarious?