= Solution
Use the printed numbering: site 1 is at the top on $+y$, sites 2 and 3 are on the right, site 4 is at the bottom, and sites 5 and 6 are on the left; $z$ is perpendicular to the ring. Let $\chi_j$ be consistently phased, <orthonormal> $p_z$ <atomic orbitals> at those sites.
In the <Hückel method>, $H_{jj}=\alpha$ and $H_{j,j\pm1}=\beta$, including the 6–1 bond. Set $\lambda=(\varepsilon-\alpha)/\beta$, with the usual $\beta<0$. The cyclic nearest-neighbour <matrix> has <eigenvalues> $\lambda_k=2\cos(2\pi k/6)$. One real, normalized <orthonormal eigenbasis> is
$$
\begin{array}{c|c|c|c}
&\text{normalized coefficients on sites }1,\ldots,6&\varepsilon&\text{electrons}\\\hline
\psi_0&(1,1,1,1,1,1)/\sqrt6&\alpha+2\beta&2\\
\psi_{c1}&(2,1,-1,-2,-1,1)/\sqrt{12}&\alpha+\beta&2\\
\psi_{s1}&(0,1,1,0,-1,-1)/2&\alpha+\beta&2\\
\psi_{c2}&(2,-1,-1,2,-1,-1)/\sqrt{12}&\alpha-\beta&0\\
\psi_{s2}&(0,1,-1,0,1,-1)/2&\alpha-\beta&0\\
\psi_3&(1,-1,1,-1,1,-1)/\sqrt6&\alpha-2\beta&0
\end{array}
$$
Each row means $\psi=\sum_jc_j\chi_j$. The three lowest <molecular orbitals> hold the six $\pi$ <electrons>. Any real <orthogonal> mixing within either degenerate pair is an equally valid choice of <molecular orbitals>; the coefficient vectors specify exactly which sketches are used here.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-1-benzene-orbitals.png]
{title=Original benzene pi-orbital energy diagram and six real orbital phase sketches in the printed atom numbering}
The coloured circles show the signs and relative magnitudes of the lobes above the molecular plane. The lobes below it have the opposite signs; colours are <quantum phases>, not <electric charges>.
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