The Perfect Reasoning

Let the numbers be $a,b,c$ from left to right.

First, note that $a \le 3$ ; if $a \ge 4$ , then $a+b+c \ge 4+5+6 = 15$ , contradiction. Likewise, $c \ge 6$ , otherwise $a+b+c \le 3+4+5 = 12$ , and $c \le 10$ , otherwise $a+b+c \ge 1+2+11 = 14$ .

Casey's statement means $a \neq 3$ (otherwise Casey would be able to deduce that $(a,b,c) = (3,4,6)$ ).

Tracy's statement on its own means $c \neq 9, 10$ (otherwise Tracy would be able to deduce that $(a,b,c) = (1,3,9), (1,2,10)$ respectively). Since Tracy's statement follows Casey's, we also know $c \neq 6$ ; if $c = 6$ then $(a,b,c) = (2,5,6), (3,4,6)$ , and since Casey says $a \neq 3$ , Tracy would know that $(a,b,c) = (2,5,6)$ .

The remaining possibilities are $a = 1,2$ with $c = 7,8$ , giving $(a,b,c) = (1,5,7), (1,4,8), (2,4,7), (2,3,8)$ . Since Stacy can't figure out the numbers even by knowing the value of $b$ , we know that there must be more than one possible $(a,b,c)$ ; the only satisfying $b$ is $\boxed{4}$ .