nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square.
 
[i]Proposed by Mihir Singhal and Michael Kural[/i]

1. **Understanding the Problem:**
   We need to find all real numbers \( d > 0 \) such that for every positive integer \( n \), we can label at least \( dn^2 \) cells of an \( n \times n \) square grid with kappas, under the constraint that no three consecutive cells (horizontally, vertically, or diagonally) contain kappas.

2. **Analyzing a \( 3 \times 3 \) Grid:**
   Let's start by considering a \( 3 \times 3 \) grid. We need to determine the maximum number of kappas that can be placed in this grid without violating the given constraints.
   
   - If we place kappas in the cells (1,1), (1,3), (2,2), (3,1), and (3,3), we have 5 kappas and no three consecutive cells in any direction contain kappas.
   - It is impossible to place more than 6 kappas in a \( 3 \times 3 \) grid without violating the constraints. Therefore, the maximum number of kappas in a \( 3 \times 3 \) grid is 6.

3. **Generalizing to Larger Grids:**
   Consider a larger grid, say \( 6 \times 6 \). We can divide this grid into four \( 3 \times 3 \) subgrids. If we place kappas optimally in each \( 3 \times 3 \) subgrid, we can place at most 6 kappas in each subgrid.

   - Therefore, in a \( 6 \times 6 \) grid, we can place at most \( 4 \times 6 = 24 \) kappas.
   - However, if we try to place more than 20 kappas (say 21), at least one of the \( 3 \times 3 \) subgrids must contain more than 5 kappas, which is not possible without violating the constraints.

4. **Calculating the Density \( d \):**
   - For a \( 3 \times 3 \) grid, the density of kappas is \( \frac{6}{9} = \frac{2}{3} \).
   - For a \( 6 \times 6 \) grid, the density of kappas is \( \frac{24}{36} = \frac{2}{3} \).

   This suggests that the maximum density \( d \) is \( \frac{2}{3} \).

5. **Conclusion:**
   Since we can place kappas in such a way that the density is \( \frac{2}{3} \) for any \( n \times n \) grid, and we cannot exceed this density without violating the constraints, we conclude that the maximum value of \( d \) is \( \frac{2}{3} \).

The final answer is \( \boxed{ d = \frac{2}{3} } \).