Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board. 

How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles?

Proposed by David Torres Flores
1. **Understanding the Problem:**
   We need to determine the number of different white figures that can be created on an \( n \times n \) board by drawing \( k \) rectangles, each covering the top-right corner of the board. The white figure should be such that it cannot be created with fewer than \( k \) rectangles.

2. **Analyzing the Rectangles:**
   Each rectangle has its top-right corner at the top-right corner of the board. The bottom-left corner of each rectangle can be any point on the board. Therefore, each rectangle can be uniquely identified by its bottom-left corner coordinates \((i, j)\) where \(1 \leq i, j \leq n\).

3. **Determining the White Region:**
   The white region left after drawing \( k \) rectangles is bounded by the bottom-left corners of these rectangles. Specifically, the white region is determined by the minimum \( k \) vertical and horizontal lines that are not covered by any of the \( k \) rectangles.

4. **Counting the Lines:**
   The white region is uniquely determined by the \( k \) vertical and \( k \) horizontal lines that are not covered by the rectangles. We need to choose \( k \) lines from the \( n \) available lines (excluding the rightmost and bottommost lines).

5. **Combinatorial Calculation:**
   The number of ways to choose \( k \) lines from \( n \) available lines is given by the binomial coefficient \(\binom{n}{k}\). Since we need to choose \( k \) vertical lines and \( k \) horizontal lines independently, the total number of ways to choose these lines is \(\binom{n}{k} \times \binom{n}{k}\).

6. **Final Expression:**
   The total number of different white figures that can be created with \( k \) rectangles is given by:
   \[
   \left( \binom{n}{k} \right)^2
   \]

The final answer is \(\boxed{\left( \binom{n}{k} \right)^2}\).