A maximum path

The label A maximum path is not enough to invent a recurrence safely. Ask whether moves are right/down, diagonally down, four directions, or whether the path must start and end at specified borders; also ask whether cells may be negative and whether revisits are allowed. For the common acyclic grid form, define dp[r][c] as the maximum score reaching that cell, initialize all states to negative infinity, set the valid start state, and relax it from each permitted predecessor. If movement is only from a previous row, compute one row at a time and retain the best answer in the final row. If arbitrary four-direction movement or revisits are allowed, ordinary DP may be invalid and the task may become a graph longest-path problem with cycle concerns. Explain this distinction rather than guessing. Test a one-cell grid, blocked or unreachable cells, negative scores, and ties. After rules are fixed, complexity is typically O(rowscolsmoves) with O(rows*cols) space or O(cols) rolling space.