Alex wants to cover a 30 by 30 board perfectly with 450 1 by 2 dominos. He also wants to ensure that he can trace a path between any 2 dominos that connect through at most N dominos. What is the minimum possible value of
N
which would allow Alex to form such a configuration
To find the minimum possible value of N that would allow Alex to form the desired configuration, we need to consider the arrangement of the dominos on the board.
Since Alex wants to cover a 30 by 30 board with 450 1 by 2 dominos, we have a total of 900 squares on the board. Each 1 by 2 domino covers 2 squares, so the dominos will cover a total of 900 * 2 = 1800 squares.
Now, let's assume that we have a configuration where Alex can trace a path between any 2 dominos connecting through at most N dominos. In such a configuration, every square on the board should be reachable by the path.
To minimize N, we need to ensure that the path can reach the furthest dominos with the minimum number of intermediate dominos. In other words, we want to minimize the maximum distance between any two dominos.
Since each domino occupies 2 squares, the maximum distance between two dominos is the Manhattan distance between their centers. In a grid with dimensions 30 by 30, the maximum Manhattan distance between any two points is 29 + 29 = 58.
However, for Alex to be able to trace a path between any two dominos connecting through at most N dominos, the maximum distance between any two dominos should be less than or equal to N.
Therefore, in this case, the minimum possible value of N is 58.