Question

Can you generalize how many ways there are to win if 3 in a row is the requirement? Let n be the square size.

a) n-2
b) 2^n
c) n+1
d) 3n

Answer 1

There are 2n + 2 ways to win on an n x n square board when requiring 3 in a row, considering the horizontal, vertical, and diagonal possibilities. This total includes n horizontal, n vertical, and 2 diagonal winning alignments. The provided answer choice '3n' does not match exactly but comes closest if ignoring the two diagonal wins.

Step-by-step explanation:

To determine how many ways there are to win if 3 in a row is the requirement on an n x n square board, we can consider the different directions in which one can achieve 3 in a row: horizontally, vertically, and diagonally.

For a square of size n, there are:

• n ways to win horizontally (one for each row).
• n ways to win vertically (one for each column).
• 2 ways to win diagonally (one for each of the two main diagonals).

Adding these up gives n (horizontal) + n (vertical) + 2 (diagonal) = 2n + 2 ways to win in total. None of the given answer choices matches this result exactly, but if we interpret the question as ignoring the two diagonal wins, the closest answer would be 'd) 3n', if n were to represent the total ways per type (ignoring diagonals).