Mathematical Group Analysis: Identifying the Property-Free Set

To determine which sequence has no consistent pattern or property, we will examine each of the given sequences for an arithmetic progression. An arithmetic progression is defined by a sequence in which each term after the first is obtained by adding a constant difference to the previous term.

• Option 1: $6, 5, 4, 3, 2, 1$ - The differences are $-1, -1, -1, -1, -1$. This follows an arithmetic progression with a common difference of $-1$.
• Option 2: $2, 1, 0, 1, 2, 3$ - The differences are $-1, -1, +1, +1, +1$. The differences are inconsistent across the entire sequence, suggesting this does not follow a simple arithmetic progression.
• Option 3: $55, 45, 35, 30, 20, 0$ - The differences are $-10, -10, -5, -10, -20$. The differences are inconsistent, indicating no uniform arithmetic progression property is present throughout.
• Option 4: $60, 50, 40, 30, 20, 10$ - The differences are $-10, -10, -10, -10, -10$. This sequence follows an arithmetic progression with a constant difference of $-10$.

From this analysis, Option 3 does not follow a consistent arithmetic progression or property, as the differences between terms are inconsistent.

Therefore, the sequence in which there is no property is $55, 45, 35, 30, 20, 0$.