A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.
1. **Calculate the area of the board:**
   The dimensions of the board are \((2^n - 1) \times (2^n + 1)\). Therefore, the area \(A\) of the board is:
   \[
   A = (2^n - 1)(2^n + 1)
   \]
   Using the difference of squares formula, we get:
   \[
   A = 2^{2n} - 1
   \]

2. **Express the area as a sum of powers of 2:**
   The area \(2^{2n} - 1\) can be expressed in binary form as a sequence of \(2n\) ones. For example, if \(n = 2\), then \(2^{2n} - 1 = 2^4 - 1 = 15\), which is \(1111_2\). This means that the number \(2^{2n} - 1\) requires \(2n\) summands when expressed as a sum of powers of 2.

3. **Determine the minimum number of rectangles:**
   Each rectangle must have an area that is a power of 2. Since the sum of the areas of the rectangles must equal \(2^{2n} - 1\), and since \(2^{2n} - 1\) requires \(2n\) summands in its binary representation, the minimum number of rectangles needed is \(2n\).

4. **Verify the construction:**
   To verify, we can construct the board by dividing it into rectangles with areas that are powers of 2. For example, we can split the rows into one row and a block of \(2^n\) rows, and then divide the \(2^n - 1\) columns into parts of lengths that are powers of 2. This ensures that each rectangle has an area that is a power of 2.

\(\blacksquare\)

The final answer is \( \boxed{ 2n } \)