MathDB

Given an

n \times n

board which is divided into

n^2

squares of size

1 \times 1

, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one

1\times 2

domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a

2\times 1

domino.
After Aqua colors, it turns out there are exactly

2024

ways for Ruby to place a domino on the board so that it covers exactly

1

black square and

1

white square.
Determine the smallest possible value of

n

so that Aqua and Ruby can do this.
Proposed by Muhammad Afifurrahman, Indonesia

2