12.4.1 Various Olympiad problems

Problem: Geppetto and Pinocchio play according to the following rules: Geppetto writes on the board six different numbers in a row, and Pinocchio comes up with his own four numbers for them \( x_{1}, x_{2}, x_{3}, x_{4} \) and under each number Geppetto writes respectively any of the sums \( x_{1}+x_{2}, x_{1}+x_{3}, x_{1}+ \) \( x_{4}, x_{2}+x_{3}, x_{2}+x_{4}, x_{3}+x_{4} \) (each only once), after which for each sum, equal to the number under it, Pinocchio gets 3 apples, and for a larger one 1 apple. What is the maximum number of apples Pinocchio can be guaranteed to get?