assign variables $a_{ij}, b_{ij}, c_{ij}, d_{ij}, e_{ij}, f_{ij}$ for the quadrants, left to right and top to bottom, $i, j \in {1..5}$.
$$
\begin{matrix}
\forall i,j: a_{ij} \in {0,1} \ & \forall i,j: b_{ij} \in {0,1} \ & \forall i,j: c_{ij} \in {0,1} \\
\forall i,j: d_{ij} \in {0,1} \ & \forall i,j: e_{ij} \in {0,1} \ & \\
\forall i,j: f_{ij} \in {0,1} \ & & \\
\end{matrix}
$$
$$
\begin{matrix}
\forall i: \sum_j a_{ij} = 1 \ & \forall i: \sum_j b_{ij} = 1 \ & \forall i: \sum_j c_{ij} = 1 \\
\forall j: \sum_i a_{ij} = 1 \ & \forall j: \sum_i b_{ij} = 1 \ & \forall j: \sum_i c_{ij} = 1 \\
\forall i: \sum_j d_{ij} = 1 \ & \forall i: \sum_j e_{ij} = 1 \ & \\
\forall j: \sum_i d_{ij} = 1 \ & \forall j: \sum_i e_{ij} = 1 \ & \\
\forall i: \sum_j f_{ij} = 1 \ & & \\
\forall j: \sum_i f_{ij} = 1 \ & & \\
\end{matrix}
$$
$$
\begin{matrix}
& \sum_j j \cdot d_{1j} - \sum_j j \cdot a_{5j} = 2 & \quad \text{stop by Will's two before the sheep dog.} \\
& e_{11} = 1 & \quad \text{Bruce is the sheep dog.} \\
& \sum_j j \cdot d_{2j} - \sum_j j \cdot a_{1j} = 1 & \quad \text{stop by Adam's one before the border collie.} \\
& \sum_j j \cdot f_{5j} - \sum_j j \cdot a_{3j} = 2 & \quad \text{stop by Dora's two before Rocky.} \\
& \sum_j j \cdot a_{5j} - \sum_j j \cdot f_{4j} = 2 & \quad \text{stop by Will's two after Wuff.} \\
& \sum_j j \cdot a_{1j} - \sum_j j \cdot d_{4j} = 1 & \quad \text{stop by Adam's one after the pit bull.} \\
& \sum_j j \cdot a_{2j} - \sum_j j \cdot f_{1j} \ge 0 & \quad \text{stop by Arthur's no later than Bruce.} \\
& \sum_j j \cdot f_{5j} \neq 2 & \quad \text{Roxie is not number 2.}\\
\end{matrix}
$$
the puzzle being mentioned: https://www.reddit.com/r/theydidthemath/comments/1ppuha4