The particle in a two-dimensional box model and graphene
The key to answering this is to remember the following principles:

1) electrons are fermions and therefore two electrons cannot share all quantum numbers (degeneracy of all QM numbers is forbidden)

2) electrons have spin quantum number, which means an energy state corresponding to a given set of $n_1$ and $n_2$ quantum numbers can be doubly occupied (doubly degenerate)

3) the Aufbau principle, which means lowest energy states are occupied first

Once you know the total number of electrons, you have to assign each a pair of quantum numbers $n_1$ and $n_2$ (ignoring $n_s$, the spin quantum #). The first two principles listed above mean that to each $(n_1,n_2)$ state you can assign two and only two electrons (with opposite spin quantum #). The third principle means that, since the energy increases with $n_i$, and the sides of the box are identical (making states that permute the $n_i$ degenerate), you should assign the $n_i$ in increasing order of energy as follows: the lowest E state has $n_1=1,n_2=1$, or (1,1); followed by (2,1) or (1,2) - these are degenerate in energy; then (2,2); then (3,1) or (1,3) -again degenerate; then (2,3) and so on. 

In tabular form the energies increase as follows:

$$\begin{array}{c|c|c|c}\hline i &amp; n_1 &amp; n_2 &amp; n_s  \\ \hline 1  &amp; 1 &amp; 1 &amp; \alpha \\   &amp; 1 &amp; 1 &amp; \beta \\ 2  &amp; 2 &amp; 1 &amp; \alpha \\   &amp; 2 &amp; 1 &amp; \beta  \\   &amp; 1 &amp; 2 &amp; \alpha \\   &amp; 1 &amp; 2 &amp; \beta  \\ 3  &amp; 2 &amp; 2 &amp; \alpha  \\   &amp; 2 &amp; 2 &amp; \beta\\ 4  &amp; 1 &amp; 3 &amp; \alpha \\   ...&amp;...&amp;...&amp; \\ \hline\end{array} $$

If you counted 288 electrons you need 144 states that are doubly degenerate (permutations of $n_i$, differing only in $n_s$) to populate the box. The HOMO is the occupied state with highest energy. 

Using a script to crack the problem (rather than searching for a mathematical formula that encodes the pattern) produced the following list of states (each can be populated by two electrons, one with $n_s=\alpha$, the other $n_s=\beta$). The HOMO is triply degenerate (ignoring spin) and the LUMO doubly degenerate.

$$\begin{array}{c|c|c|c}\hline i &amp; n_1^2 +  n_2^2 &amp; n_1 &amp; n_2  \\ \hline 1 &amp; 2 &amp; 1 &amp; 1 \\ 
     2 &amp; 5 &amp; 1 &amp; 2 \\ 
     3 &amp; 5 &amp; 2 &amp; 1 \\ 
     4 &amp; 8 &amp; 2 &amp; 2 \\ 
     5 &amp; 10 &amp; 1 &amp; 3 \\ 
     6 &amp; 10 &amp; 3 &amp; 1 \\ 
     7 &amp; 13 &amp; 2 &amp; 3 \\ 
     8 &amp; 13 &amp; 3 &amp; 2 \\ 
     9 &amp; 17 &amp; 1 &amp; 4 \\ 
     10 &amp; 17 &amp; 4 &amp; 1 \\ 
     11 &amp; 18 &amp; 3 &amp; 3 \\ 
     12 &amp; 20 &amp; 2 &amp; 4 \\ 
     13 &amp; 20 &amp; 4 &amp; 2 \\ 
     14 &amp; 25 &amp; 3 &amp; 4 \\ 
     15 &amp; 25 &amp; 4 &amp; 3 \\ 
     16 &amp; 26 &amp; 1 &amp; 5 \\ 
     17 &amp; 26 &amp; 5 &amp; 1 \\ 
     18 &amp; 29 &amp; 2 &amp; 5 \\ 
     19 &amp; 29 &amp; 5 &amp; 2 \\ 
     20 &amp; 32 &amp; 4 &amp; 4 \\ 
     21 &amp; 34 &amp; 3 &amp; 5 \\ 
     22 &amp; 34 &amp; 5 &amp; 3 \\ 
     23 &amp; 37 &amp; 1 &amp; 6 \\ 
     24 &amp; 37 &amp; 6 &amp; 1 \\ 
     25 &amp; 40 &amp; 2 &amp; 6 \\ 
     26 &amp; 40 &amp; 6 &amp; 2 \\ 
     27 &amp; 41 &amp; 4 &amp; 5 \\ 
     28 &amp; 41 &amp; 5 &amp; 4 \\ 
     29 &amp; 45 &amp; 3 &amp; 6 \\ 
     30 &amp; 45 &amp; 6 &amp; 3 \\ 
     31 &amp; 50 &amp; 1 &amp; 7 \\ 
     32 &amp; 50 &amp; 5 &amp; 5 \\ 
     33 &amp; 50 &amp; 7 &amp; 1 \\ 
     34 &amp; 52 &amp; 4 &amp; 6 \\ 
     35 &amp; 52 &amp; 6 &amp; 4 \\ 
     36 &amp; 53 &amp; 2 &amp; 7 \\ 
     37 &amp; 53 &amp; 7 &amp; 2 \\ 
     38 &amp; 58 &amp; 3 &amp; 7 \\ 
     39 &amp; 58 &amp; 7 &amp; 3 \\ 
     40 &amp; 61 &amp; 5 &amp; 6 \\ 
     41 &amp; 61 &amp; 6 &amp; 5 \\ 
     42 &amp; 65 &amp; 1 &amp; 8 \\ 
     43 &amp; 65 &amp; 4 &amp; 7 \\ 
     44 &amp; 65 &amp; 7 &amp; 4 \\ 
     45 &amp; 65 &amp; 8 &amp; 1 \\ 
     46 &amp; 68 &amp; 2 &amp; 8 \\ 
     47 &amp; 68 &amp; 8 &amp; 2 \\ 
     48 &amp; 72 &amp; 6 &amp; 6 \\ 
     49 &amp; 73 &amp; 3 &amp; 8 \\ 
     50 &amp; 73 &amp; 8 &amp; 3 \\ 
     51 &amp; 74 &amp; 5 &amp; 7 \\ 
     52 &amp; 74 &amp; 7 &amp; 5 \\ 
     53 &amp; 80 &amp; 4 &amp; 8 \\ 
     54 &amp; 80 &amp; 8 &amp; 4 \\ 
     55 &amp; 82 &amp; 1 &amp; 9 \\ 
     56 &amp; 82 &amp; 9 &amp; 1 \\ 
     57 &amp; 85 &amp; 2 &amp; 9 \\ 
     58 &amp; 85 &amp; 6 &amp; 7 \\ 
     59 &amp; 85 &amp; 7 &amp; 6 \\ 
     60 &amp; 85 &amp; 9 &amp; 2 \\ 
     61 &amp; 89 &amp; 5 &amp; 8 \\ 
     62 &amp; 89 &amp; 8 &amp; 5 \\ 
     63 &amp; 90 &amp; 3 &amp; 9 \\ 
     64 &amp; 90 &amp; 9 &amp; 3 \\ 
     65 &amp; 97 &amp; 4 &amp; 9 \\ 
     66 &amp; 97 &amp; 9 &amp; 4 \\ 
     67 &amp; 98 &amp; 7 &amp; 7 \\ 
     68 &amp; 100 &amp; 6 &amp; 8 \\ 
     69 &amp; 100 &amp; 8 &amp; 6 \\ 
     70 &amp; 101 &amp; 1 &amp; 10 \\ 
     71 &amp; 101 &amp; 10 &amp; 1 \\ 
     72 &amp; 104 &amp; 2 &amp; 10 \\ 
     73 &amp; 104 &amp; 10 &amp; 2 \\ 
     74 &amp; 106 &amp; 5 &amp; 9 \\ 
     75 &amp; 106 &amp; 9 &amp; 5 \\ 
     76 &amp; 109 &amp; 3 &amp; 10 \\ 
     77 &amp; 109 &amp; 10 &amp; 3 \\ 
     78 &amp; 113 &amp; 7 &amp; 8 \\ 
     79 &amp; 113 &amp; 8 &amp; 7 \\ 
     80 &amp; 116 &amp; 4 &amp; 10 \\ 
     81 &amp; 116 &amp; 10 &amp; 4 \\ 
     82 &amp; 117 &amp; 6 &amp; 9 \\ 
     83 &amp; 117 &amp; 9 &amp; 6 \\ 
     84 &amp; 122 &amp; 1 &amp; 11 \\ 
     85 &amp; 122 &amp; 11 &amp; 1 \\ 
     86 &amp; 125 &amp; 2 &amp; 11 \\ 
     87 &amp; 125 &amp; 5 &amp; 10 \\ 
     88 &amp; 125 &amp; 10 &amp; 5 \\ 
     89 &amp; 125 &amp; 11 &amp; 2 \\ 
     90 &amp; 128 &amp; 8 &amp; 8 \\ 
     91 &amp; 130 &amp; 3 &amp; 11 \\ 
     92 &amp; 130 &amp; 7 &amp; 9 \\ 
     93 &amp; 130 &amp; 9 &amp; 7 \\ 
     94 &amp; 130 &amp; 11 &amp; 3 \\ 
     95 &amp; 136 &amp; 6 &amp; 10 \\ 
     96 &amp; 136 &amp; 10 &amp; 6 \\ 
     97 &amp; 137 &amp; 4 &amp; 11 \\ 
     98 &amp; 137 &amp; 11 &amp; 4 \\ 
     99 &amp; 145 &amp; 1 &amp; 12 \\ 
     100 &amp; 145 &amp; 8 &amp; 9 \\ 
     101 &amp; 145 &amp; 9 &amp; 8 \\ 
     102 &amp; 145 &amp; 12 &amp; 1 \\ 
     103 &amp; 146 &amp; 5 &amp; 11 \\ 
     104 &amp; 146 &amp; 11 &amp; 5 \\ 
     105 &amp; 148 &amp; 2 &amp; 12 \\ 
     106 &amp; 148 &amp; 12 &amp; 2 \\ 
     107 &amp; 149 &amp; 7 &amp; 10 \\ 
     108 &amp; 149 &amp; 10 &amp; 7 \\ 
     109 &amp; 153 &amp; 3 &amp; 12 \\ 
     110 &amp; 153 &amp; 12 &amp; 3 \\ 
     111 &amp; 157 &amp; 6 &amp; 11 \\ 
     112 &amp; 157 &amp; 11 &amp; 6 \\ 
     113 &amp; 160 &amp; 4 &amp; 12 \\ 
     114 &amp; 160 &amp; 12 &amp; 4 \\ 
     115 &amp; 162 &amp; 9 &amp; 9 \\ 
     116 &amp; 164 &amp; 8 &amp; 10 \\ 
     117 &amp; 164 &amp; 10 &amp; 8 \\ 
     118 &amp; 169 &amp; 5 &amp; 12 \\ 
     119 &amp; 169 &amp; 12 &amp; 5 \\ 
     120 &amp; 170 &amp; 1 &amp; 13 \\ 
     121 &amp; 170 &amp; 7 &amp; 11 \\ 
     122 &amp; 170 &amp; 11 &amp; 7 \\ 
     123 &amp; 170 &amp; 13 &amp; 1 \\ 
     124 &amp; 173 &amp; 2 &amp; 13 \\ 
     125 &amp; 173 &amp; 13 &amp; 2 \\ 
     126 &amp; 178 &amp; 3 &amp; 13 \\ 
     127 &amp; 178 &amp; 13 &amp; 3 \\ 
     128 &amp; 180 &amp; 6 &amp; 12 \\ 
     129 &amp; 180 &amp; 12 &amp; 6 \\ 
     130 &amp; 181 &amp; 9 &amp; 10 \\ 
     131 &amp; 181 &amp; 10 &amp; 9 \\ 
     132 &amp; 185 &amp; 4 &amp; 13 \\ 
     133 &amp; 185 &amp; 8 &amp; 11 \\ 
     134 &amp; 185 &amp; 11 &amp; 8 \\ 
     135 &amp; 185 &amp; 13 &amp; 4 \\ 
     136 &amp; 193 &amp; 7 &amp; 12 \\ 
     137 &amp; 193 &amp; 12 &amp; 7 \\ 
     138 &amp; 194 &amp; 5 &amp; 13 \\ 
     139 &amp; 194 &amp; 13 &amp; 5 \\ 
     140 &amp; 197 &amp; 1 &amp; 14 \\ 
     141 &amp; 197 &amp; 14 &amp; 1 \\ 
     \text{HOMO}\,142 &amp; 200 &amp; 2 &amp; 14 \\ 
     \text{HOMO}\,143 &amp; 200 &amp; 10 &amp; 10 \\ 
    \text{HOMO}\, 144 &amp; 200 &amp; 14 &amp; 2  \\ 
     \text{LUMO}\, 145 &amp; 202 &amp; 9 &amp; 11 \\ 
    \text{LUMO}\, 146 &amp; 202 &amp; 11 &amp; 9 \\ 
     147 &amp; 205 &amp; 3 &amp; 14 \\ 
     148 &amp; 205 &amp; 6 &amp; 13 \\ 
     149 &amp; 205 &amp; 13 &amp; 6 \\ 
     150 &amp; 205 &amp; 14 &amp; 3 \\ 
     151 &amp; 208 &amp; 8 &amp; 12 \\ 
     152 &amp; 208 &amp; 12 &amp; 8 \\ 
     153 &amp; 212 &amp; 4 &amp; 14 \\ 
     154 &amp; 212 &amp; 14 &amp; 4 \\ 
     155 &amp; 218 &amp; 7 &amp; 13 \\ 
     156 &amp; 218 &amp; 13 &amp; 7 \\ 
     157 &amp; 221 &amp; 5 &amp; 14 \\ 
     158 &amp; 221 &amp; 10 &amp; 11 \\ 
     159 &amp; 221 &amp; 11 &amp; 10 \\ 
     160 &amp; 221 &amp; 14 &amp; 5 \\ \hline\end{array} $$