6.29 For any real number $x$, the function $f(x)$ has the property
$$f(x)+f(x-1)=x^{2}$$

If $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?
[Solution] From the property of $f(x)$, we get
$$f(x)=x^{2}-f(x-1) .$$

Repeatedly applying the above formula, we obtain
$$\begin{aligned}
f(94) & =94^{2}-f(93) \\
& =94^{2}-93^{2}+f(92) \\
& =94^{2}-93^{2}+92^{2}-f(91) \\
& \cdots \\
& =94^{2}-93^{2}+92^{2}-91^{2}+\cdots+20^{2}-f(19) \\
& =187+183+\cdots+43+400-94 \\
& =\frac{187+43}{2} \cdot\left(\frac{187-43}{4}+1\right)+400-94 \\
& =4561 .
\end{aligned}$$

Therefore, the remainder when $f(94)$ is divided by 1000 is 561.