G3.4 There are 4 problems in a mathematics competition. The scores of each problem are allocated in the following ways: 2 marks will be given for a correct answer, 1 mark will be deducted from a wrong answer and 0 mark will be given for a blank answer. To ensure that 3 candidates will have the same scores, there should be at least $S$ candidates in the competition. Find the value of $S$.
We shall tabulate different cases:
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||c|}
\hline case no. & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & marks for each question \\
\hline correct & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 \\
\hline blank & 0 & 1 & 0 & 2 & 0 & 1 & 3 & 2 & 1 & 1 & 4 & 3 & 2 & 1 & 0 & 0 \\
\hline wrong & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 & 4 & -1 \\
\hline Total & 8 & 6 & 5 & 4 & 2 & 3 & 2 & 1 & 0 & -1 & 0 & -1 & -2 & -3 & -4 & \\
\hline
\end{tabular}

The possible total marks for one candidate to answer 4 questions are: $8,6,5,4,3,2,1,0,-1,-2,-3,-4$; altogether 12 possible combinations.
To ensure 3 candidates will have the same scores, we consider the worst scenario:
Given that there are 24 candidates. 2 candidates score 8 marks, 2 candidates score 6 marks, $\qquad$ 2 candidates score -4 marks, then the $25^{\text {th }}$ candidate will score the same as the other two candidates.