Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$ on the board. Then, he told Gauss:

On every turn, erase two random numbers $a, b$, chosen uniformly randomly, from the board and then write the term $2ab - a - b + 1$ on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?
1. **Define the problem and the initial setup:**
   We are given 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$, where $\zeta_{15} = e^{2\pi i/15}$. We need to find the expected value of the last number remaining after repeatedly applying the operation $2ab - a - b + 1$.

2. **Understand the operation:**
   The operation $2ab - a - b + 1$ can be rewritten as:
   \[
   2ab - a - b + 1 = (2a-1)(2b-1)/2 + 1/2
   \]
   This suggests a transformation of the form $x \to 2x - 1$.

3. **Transform the initial numbers:**
   Let $x_k = 2\zeta_{15}^k - 1$. The operation then becomes:
   \[
   2ab - a - b + 1 = \frac{(2a-1)(2b-1)}{2} + \frac{1}{2}
   \]
   In terms of $x_k$, this is:
   \[
   \frac{x_k x_j}{2} + \frac{1}{2}
   \]

4. **Simplify the product:**
   We need to find the product of all $x_k$:
   \[
   \prod_{k=0}^{14} x_k = \prod_{k=0}^{14} (2\zeta_{15}^k - 1)
   \]
   Notice that $\zeta_{15}^k$ are the 15th roots of unity, and their product is $(-1)^{15-1} = -1$.

5. **Calculate the product:**
   \[
   \prod_{k=0}^{14} (2\zeta_{15}^k - 1) = 2^{15} \prod_{k=0}^{14} \left(\zeta_{15}^k - \frac{1}{2}\right)
   \]
   Since $\zeta_{15}^k$ are roots of unity, the product $\prod_{k=0}^{14} \zeta_{15}^k = (-1)^{15-1} = -1$.

6. **Evaluate the final expression:**
   \[
   \prod_{k=0}^{14} \left(\zeta_{15}^k - \frac{1}{2}\right) = \left(\frac{1}{2}\right)^{15} - 1 = -\frac{1}{2^{15}}
   \]
   Therefore:
   \[
   2^{15} \left(-\frac{1}{2^{15}}\right) = -1
   \]

7. **Final transformation:**
   The final number is:
   \[
   \frac{-1}{2} + \frac{1}{2} = 0
   \]

The final answer is $\boxed{0}$