Out of 4 statements, exactly $3$ are false and $1$ is true.
- If statement 4 is true then it implies statement 3 is also true (but only 1 statement can be true).
- If statement 3 is true then it implies statement 2 is also true (again only 1 statement can be true).
- If statement 2 is true then all other statements are false including statement 3 which states "Chacko says that Binky is telling the truth". This means Binky is not telling the truth, so statement 2 can not be true.
Thus statements $2,3$ and $4$ cannot be possibly true implying statement $1$ is true.
Statement 1 states "Anuj says that Binky did it", so Binky is the culprit.