Problem: Place the 2004 positive integers $1, 2, \cdots, 2004$ randomly on a circle. By counting the parity of all adjacent 3 numbers, it is known that there are 600 groups where all 3 numbers are odd, and 500 groups where exactly 2 numbers are odd. How many groups have exactly 1 odd number? How many groups have no odd numbers at all?
(2004, Shanghai TI Cup High School Mathematics Competition for Grade 10)
Solution: These 2004 numbers are arbitrarily placed on a circle, with every 3 adjacent numbers forming a group, making a total of 2004 groups. Each number is part of 3 different groups. Let the number of groups with exactly 1 odd number be $x$, and the number of groups with no odd numbers be $y$. Considering the odd number scenarios, we have
$$
600 \times 3 + 500 \times 2 + x = 1002 \times 3 .
$$

Solving for $x$ gives $x=206$.
Thus, $y=2004-600-500-206=698$.