How many four digit numbers divisible by 29 have the sum of their digits 29?

If the sum of the digits of $n$ is $29$, then the number $n$ must be congruent to $2$ (modulo $9$). Since $29$ and $9$ are relatively prime, and their product is $261$, we need only consider numbers congruent to $29$ (modulo $261$). There are only about three dozen candidates between $1000$ and $9999$.

Furthermore, the average of the digits is $\frac{29}4>7$; no digit can be $1$, and only one digit can be below $6$. That means we can start our search at $2999$; the first candidate is $3161$, easily discarded, and we just repeatedly add $261$ until we get above $9999$.