Final answer:
Without specific details about the hexagon layout and movement rules, it is not possible to accurately determine the number of ways a game piece can be moved from hexagon a to hexagon b in four moves.
Step-by-step explanation:
The question 'How many different ways can a game piece be moved from hexagon a to hexagon b in just four moves?' requires the use of combinatorial mathematics to determine the number of distinct paths that can be taken given certain constraints. The question does not provide enough specific information about the layout of the hexagons or the rules of movement for a definitive answer.
However, assuming each move must go to an adjacent hexagon and that the moves are within a regular hexagonal grid, we could envision moving in any of the six directions, and then counting the permutations of the moves. But the lack of a specific diagram or more detailed rules means we cannot give a specific number of ways to move the game piece. Combinatorics often involves factorial calculations, such as the formula for permutations 4! (four factorial), which is 4×3×2×1, but this cannot be applied without more information.
If the student can provide further details about the restrictions and layout, a more precise answer involving combinations and permutations might be possible.