Count fillings of a 3x3 grid where every 2x2 block holds 1 to 4

Each cell of a 3×3 grid is filled with one of the numbers 1, 2, 3 and 4. The grid contains four 2×2 blocks of adjacent cells: top-left, top-right, bottom-left and bottom-right. Neighbouring blocks overlap, and all four share the centre cell. In how many ways can the grid be filled so that each of the four 2×2 blocks contains each of 1, 2, 3 and 4 exactly once? The question comes from a math assessment taken on camera without a calculator, in which each question allowed five minutes to solve and five minutes to explain the reasoning. The report gives only this question's answer and reasoning; the statement above is reconstructed from that reasoning. ```hint Fix one block first Fill one 2×2 block completely, then ask how much freedom each block that overlaps it has left. ``` ```hint Check the last block Once the blocks next to your first block are filled, the remaining block is already mostly determined. Check whether it can end up with a repeated number. ``` ### Constraints and Clarifications - Only the four 2×2 blocks of adjacent cells are constrained. Other groups of four cells, such as the four corners, may repeat a number. - Two fillings are different when any cell differs, so fillings that are rotations or reflections of each other are counted separately. - An exact count is expected, with a justification short enough to explain in five minutes. ### Clarifying Questions - Must other groups of four cells, such as the four corners, also contain different numbers? - Do fillings that are rotations or reflections of each other count once or separately? ### What a Strong Answer Covers - A clear picture of how the four blocks overlap: which cells each block shares with the others - A systematic count that handles the constraint the last block places on the choices made for the others - The exact count, confirmed by a second, independent argument - A concise explanation that fits the five-minute format ### Follow-up Questions - If each of the nine cells were filled with 1, 2, 3 or 4 uniformly at random, what is the probability that the grid satisfies the condition? - How does the count change if each cell may hold any of 1 to 5 and each 2×2 block must contain four different numbers? - If fillings related by a rotation or reflection of the grid are counted once, how many essentially different fillings are there?