The pattern is such that there will be 9 single digits (1-9), then 180 2-digit numbers (10-99), because there are 90 numbers between 10 and 99. So there are 189 digits by the time we get to the first 3-digit number (100). That leaves 11 more digits after 99:
| DIGIT NUMBER |
190 |
191 |
192 |
193 |
194 |
195 |
196 |
197 |
198 |
199 |
200 |
| DIGIT |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
2 |
1 |
0 |
Therefore, the 200th digit is zero.