Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed?
1. **Understanding the Problem**:
   We need to color \(n^2\) unit squares in the \(xy\) plane such that for any \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the squares centered at \((i,k)\), \((j,k)\), and \((j,l)\) have distinct colors.

2. **Analyzing the Condition**:
   The condition states that the three squares \((i,k)\), \((j,k)\), and \((j,l)\) must have distinct colors. This implies that for any fixed \(k\), the squares in the same row \(k\) and different columns \(i\) and \(j\) must have different colors. Similarly, for any fixed \(j\), the squares in the same column \(j\) and different rows \(k\) and \(l\) must have different colors.

3. **Coloring Strategy**:
   To satisfy the condition, we need to ensure that no two squares in the same row or column share the same color. This suggests that we need a distinct color for each diagonal of the grid.

4. **Counting the Diagonals**:
   In an \(n \times n\) grid, there are \(2n-1\) diagonals. This is because there are \(n\) diagonals running from the top-left to the bottom-right (main diagonal and those above and below it) and \(n-1\) diagonals running from the top-right to the bottom-left (excluding the main diagonal which is already counted).

5. **Assigning Colors**:
   We can assign a unique color to each of these \(2n-1\) diagonals. This ensures that no two squares in the same row or column will have the same color, thus satisfying the given condition.

6. **Conclusion**:
   Therefore, the least number of colors needed to color the \(n^2\) unit squares such that the given condition is satisfied is \(2n-1\).

\[
\boxed{2n-1}
\]