A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k$ where $a_1, \dots , a_k$ are simple.
To determine if \( k = 9 \) is the least positive integer such that each positive integer \( n \) can be written as \( n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k \) where \( a_1, \dots , a_k \) are simple, we need to explore the construction and verify if a smaller \( k \) can work.

1. **Understanding the Construction for \( k = 9 \)**:
   - Given a positive integer \( n \), we can represent it in its decimal form.
   - Construct nine simple integers \( s_1, s_2, \ldots, s_9 \) such that the \( k \)-th digit in \( s_i \) equals \( 1 \) if and only if the \( k \)-th digit of \( n \) is \( i \) or larger.
   - This ensures that each digit of \( n \) is represented by a combination of the digits of \( s_1, s_2, \ldots, s_9 \).

2. **Example for Clarity**:
   - Consider \( n = 432 \).
   - The simple integers \( s_1, s_2, \ldots, s_9 \) would be:
     \[
     s_1 = 111, \quad s_2 = 111, \quad s_3 = 111, \quad s_4 = 100, \quad s_5 = 000, \quad s_6 = 000, \quad s_7 = 000, \quad s_8 = 000, \quad s_9 = 000
     \]
   - Here, \( s_1, s_2, s_3 \) have all digits as 1 because the digits of \( n \) (4, 3, 2) are all greater than or equal to 1, 2, and 3 respectively. \( s_4 \) has a 1 in the first digit because the first digit of \( n \) is 4, which is greater than or equal to 4.

3. **Checking if \( k = 8 \) or Smaller Can Work**:
   - Suppose \( k = 8 \). We need to check if every positive integer \( n \) can be represented using 8 simple integers.
   - Consider a number \( n \) with a digit 9. To represent this digit, we need at least 9 simple integers because each simple integer can only contribute a 1 or 0 in each digit place.
   - Therefore, \( k = 8 \) is insufficient to represent a digit 9 in \( n \).

4. **Conclusion**:
   - Since \( k = 8 \) is insufficient and \( k = 9 \) works for any positive integer \( n \), \( k = 9 \) is indeed the least positive integer that satisfies the condition.

The final answer is \( \boxed{ k = 9 } \).