1.12 ** Let the set $S=\left\{y \mid y=\sum_{k=1}^{1004} x_{2 k-1} x_{2 k}\right.$, where $x_{1}, x_{2}, \cdots, x_{2008} \in\{\sqrt{2}-$ $1, \sqrt{2}+1\}\}$. How many different integers are there in $S$?
From the conditions, we know that \( x_{2 k-1} x_{2 k} \in \left\{ (\sqrt{2}-1)^{2}, (\sqrt{2}-1)(\sqrt{2}+1), (\sqrt{2}+1)^{2} \right\} = \{3-2 \sqrt{2}, 1, 3+2 \sqrt{2}\} \). Let the number of indices \( k \) such that \( x_{2 k-1} x_{2 k} = 3-2 \sqrt{2}, 1 \), and \( 3+2 \sqrt{2} \) be \( a, b, \) and \( c \) respectively. Then \( y = (3-2 \sqrt{2}) a + b + (3+2 \sqrt{2}) c \), and \( a + b + c = 1004 \). The necessary and sufficient condition for \( y \in \mathbf{Z} \) is \( a = c \), which reduces to finding the number of non-negative integer solutions to \( 2a + b = 1004 \). \( a \) can take values 0, 1, ..., 502. Therefore, there are 503 different integers in \( S \).