To solve this problem, we need to determine how Ana can place the maximum number of tokens on a 2020 × 2020 board such that no two tokens share a row, column, or diagonal. This is a classic problem of placing tokens on a board such that they are not attacking each other, similar to the n-queens problem but simplified to exclude diagonals.

Ana must place the tokens in such a way that they form a checkerboard pattern to maximize the number of tokens. Each token should be placed in alternating rows and columns. This means that if a token is placed in a particular cell, the adjacent cells (horizontally, vertically, and diagonally) cannot contain any tokens.

To determine how many tokens can be placed, consider a checkerboard pattern where tokens occupy either all the black squares or all the white squares. For an $n \times n$ board, half of the squares (approximately) can be filled in this pattern.

Solution:

1. Checkerboard Pattern Approach:

   • A checkerboard pattern on an $n \times n$ board divides the board into two equal parts of approximately $\frac{n^2}{2}$ squares each.
   • For a 2020 × 2020 board, the total number of squares is $2020^2$.

2. Calculating the Maximum Number of Tokens:

   • The maximum number of tokens Ana can place is approximately half the total number of squares on the board because of the checkerboard pattern.
   • Hence, the maximum number of tokens is: $\text{Maximum Tokens} = \left\lfloor \frac{2020^2}{2} \right\rfloor$
   • Calculating this: $2020^2 = 4080400$ $\frac{4080400}{2} = 2040200$

Therefore, Ana can place a maximum of 2,040,200 tokens on a 2020 × 2020 board, which matches option (E) $2020^2$.

The correct answer is therefore (E) 2020^2.

Would you like more details or have any questions about this problem?

Here are some related questions to expand your understanding:

1. How does the placement strategy change if the board size is an odd number, like 2021 × 2021?
2. What if the tokens can attack diagonally, how would the strategy change?
3. What is the maximum number of tokens on an 8 × 8 board using this method?
4. How does this problem relate to the n-queens problem?
5. How many tokens can be placed on a 7 × 7 board using the same strategy?
6. What is the general formula for the maximum number of tokens on an $n \times n$ board?
7. How does the number of tokens change if the board is rectangular, such as 2020 × 2021?
8. Can this method be adapted to three-dimensional boards?

Tip: In problems involving grids, consider symmetry and patterns such as checkerboards to simplify and optimize arrangements.