Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?
1. **Coloring the Chessboard:**
   We start by coloring the $8 \times 8$ chessboard in a specific pattern to help us analyze the problem. The coloring is done in such a way that each $3 \times 1$ rectangle covers exactly one square of each color (1, 2, and 3). The coloring is as follows:
   \[
   \begin{bmatrix}
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1 \\
   3 & 2 & 1 & 3 & 2 & 1 & 3 & 2 \\
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1 \\
   3 & 2 & 1 & 3 & 2 & 1 & 3 & 2 \\
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1
   \end{bmatrix}
   \]

2. **Counting the Squares:**
   In this coloring, there are 22 squares with a 1, 21 squares with a 2, and 21 squares with a 3. Since each $3 \times 1$ rectangle covers one square of each color, placing 21 rectangles will cover all 21 squares with a 2, all 21 squares with a 3, and 21 out of the 22 squares with a 1. This leaves exactly one square with a 1 uncovered.

3. **Symmetry Consideration:**
   To ensure that the solution is not dependent on the specific arrangement, we reflect the coloring matrix:
   \[
   \begin{bmatrix}
   3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 \\
   1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 \\
   2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\
   3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 \\
   1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 \\
   2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\
   3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 \\
   1 & 2 & 3 & 1 & 2 & 3 & 1 & 2
   \end{bmatrix}
   \]
   By symmetry, the same rules apply, and the leftover square must be a 1-square in both colorings.

4. **Identifying Possible Positions:**
   The only positions that satisfy this condition in both colorings are:
   \[
   \begin{bmatrix}
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1 \\
   3 & 2 & \boxed{1} & 3 & 2 & \boxed{1} & 3 & 2 \\
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1 \\
   3 & 2 & \boxed{1} & 3 & 2 & \boxed{1} & 3 & 2 \\
   1 & 3 & 2 & 1 & 3 & 2 & 1 & 3 \\
   2 & 1 & 3 & 2 & 1 & 3 & 2 & 1
   \end{bmatrix}
   \]
   These positions are (3,3), (3,6), (6,3), and (6,6).

5. **Verification by Placement:**
   We can verify that it is possible to place the $3 \times 1$ rectangles such that one of these squares is left uncovered. For example:
   \[
   \begin{bmatrix}
   1 & 2 & 3 & 3 & 3 & & & \\
   1 & 2 & 4 & 4 & 4 & & & \\
   1 & 2 & & 5 & 6 & & & \\
   7 & 7 & 7 & 5 & 6 & & & \\
   8 & 8 & 8 & 5 & 6 & & & \\
   & & & & & & & \\
   & & & & & & & \\
   & & & & & & &
   \end{bmatrix}
   \]
   The rest of the board can be filled similarly, ensuring that one of the identified squares remains uncovered.