3. Find all real number triples $(x, y, z)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
{[x]+\{y\}=z,} \\
{[y]+\{z\}=x,} \\
{[z]+\{x\}=y}
\end{array}\right.
$$

(Note: $x=[x]+\{x\},[x]$ is an integer, $0 \leqslant\{x\}<1$ ).
3. Rewrite the original system of equations as
$$
\left\{\begin{array}{l}
{[x]+\{y\}=[z]+\{z \mid .} \\
{[y]+|z|=[x]+\mid x\} . \\
{[z]+\mid x\}=[y]+|y| .
\end{array}\right.
$$

Since for any real number $x, x=[x]+\{x\}$, from equation (1) we get $[x]=[z]$, from equation (2) we get $[y]=[x]$, and from equation (3) we get $[z]=[y]$.
Therefore, $[x]=[y]=[z]=n$.
Similarly, we get $\{x\}=\{y\}=\{z\}=r$.
Thus, $x=y=z=n+r$.
Since for any $x=y=z$, we know that $x=[x]+\{x\}$ holds for each equation in the system, therefore, all real number solutions to the system are $\{(t, t, t), t \in \mathbf{R}\}$.