COINS - If 2, proves C because JOINT - 1 and WORDS - 1
PULSE - If 1, proves L because PRINT - 0, EXIST - 0, and MERGE - 0.
If both C and L are disproved, then _O_O_ is proven because COLOR - 2 and COLON - 2
The way I understood the question, we would be counting the sum of payments made rather than the net difference. In that case, any complete circuit route of 49 moves does require that at least one of them pays a total of at least $25.20.
It is possible, in 49 moves, that each pays a total of $25.20 for a net of zero with neither exceeding a net of $12.60.
"the middle-middle square is adjacent to all other square"
This implies that the restriction applies to diagonally adjacent squares. There are only two solutions, along with their vertical reflections, that qualify in this case.