2006 AMC 12B Problem 18

Each step changes the coordinate sum by $1,$ so after $10$ steps the endpoint $(a, b)$ has $a + b$ even, and $|a| + |b| \le 10.$ Any such point is reachable: walk $|a| + |b|$ steps to it, then use the remaining even number of steps going out and back. The reachable points lie on the lines $a + b = 2k$ for $-5 \le k \le 5.$ Each such line meets the diamond in exactly $11$ lattice points. With $11$ lines and $11$ points each, there are $121$ points. Thus, the correct answer is B.