Final answer:
In a 10x10 grid problem where the goal is to push a cart into the Queen's castle last for credit, the optimal strategy is to push first and maintain the cart's position on the main diagonal to guarantee the final move.
Step-by-step explanation:
The scenario presented is a mathematical problem that deals with strategy and game theory. The goal is to push a cart onto the Queen's castle and whoever does this last receives most of the credit.
The playing area is a 10x10 grid, and movements are restricted to either up or right. This creates a series of options for each player on where to move the cart. One way to analyze this is by looking at the grid as a set of pathways. Notably, every move your friend makes limits the subsequent moves you can make, and vice versa.
The optimal strategy is to ensure that you are the one to push the cart into the final square. Given that the grid is even, by pushing the cart first and positioning it along the main diagonal (from the bottom-left to top-right corner), it will guarantee you the last move. If you take the first turn and make sure to keep the cart on this diagonal line, your friend will have no choice but to eventually push the cart one space away from the castle, setting you up for the final move.
Consequently, you would want to push the cart first and always keep it along this main diagonal. Such a strategy makes it impossible for your friend to reach the castle before you, granting you most of the credit for delivering the cart.