= Solution
The tableau is the three-stage <Lobatto IIIA method>. For its stage <matrix> $A$, weights $b$, nodes $c$ and $C=\operatorname{diag}(c)$, the <Butcher order conditions> through order four are
$$
\begin{gathered}
b^T\mathbf1=1,\quad b^Tc=\frac12,\quad b^Tc^{\circ2}=\frac13,\quad b^TAc=\frac16,\\
b^Tc^{\circ3}=\frac14,\quad b^TCAc=\frac18,\quad b^TAc^{\circ2}=\frac1{12},\quad b^TA^2c=\frac1{24},
\end{gathered}
$$
where $c^{\circ j}$ means coordinatewise powers. Direct substitution verifies all eight identities. Equivalently, this is a <collocation Runge-Kutta method> at the three <Lobatto quadrature> nodes, whose weights integrate cubic <polynomials> exactly.
Order five would require $b^Tc^{\circ4}=1/5$, but here $b^Tc^{\circ4}=5/24$. Therefore the method has \b[exactly order four].
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