In the first example, a possible sequence of function calls could look like this:

Initially, the function \texttt{solve(4)} is called.

From the function \texttt{solve}, the function 

\begin{verbatim}
compare({Triangle(0, 1, 2)}, {Triangle(1, 2, 3)}) 
\end{verbatim}

is called. 

During the execution of the function, the sizes of the triangular pieces formed by the triangle with vertices at candles numbered $0$, $1$, and $2$ and the triangle with vertices at candles $1$, $2$, and $3$ are compared.

Since the first triangle is smaller than the second, the function returns $-1$.

Next, from the function \texttt{solve}, the function 

\begin{verbatim}
compare({Triangle(1, 2, 3)}, {Triangle(0, 1, 2), Triangle(0, 1, 3)}) 
\end{verbatim}

is called.

The response will return $1$, as the area of the triangle with vertices at candles $1$, $2$, and $3$ is greater than the total area of the triangle with vertices at candles $0$, $1$, and $2$ and the triangle with vertices at candles $0$, $1$, and $3$.

Next, from the function \texttt{solve}, the function 

\begin{verbatim}
compare({Triangle(1, 2, 3)}, {Triangle(0, 1, 2), Triangle(2, 3, 0), Triangle(0, 1, 3)}) 
\end{verbatim}

is called.

The response will return $0$.

In the end, the function \texttt{solve} returns \t{\{\{0, 1, 2\}, \{0, 2, 3\}, \{0, 1, 3\}\}}~--- a description of the partitioning of the cake into convex pieces that have the maximum possible total area and do not intersect at internal points.

In the first and third examples, the found method of cutting the cake has the maximum possible number of pieces, so such answers would be considered correct in groups with maximization. In the second test, the number of pieces is not maximally possible, so such an answer would be considered correct in groups without maximization, but would receive $0$ points in groups with maximization.

Note that the first test fits the additional constraints of groups $1$ and $2$, while the third test fits the additional constraints of group $5$.