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This direct approach, programmed in Python 3, runs in 5.1s. With a bit of analysis we can do better than this, but I like this approach as it doesn’t need to make assumptions to solve the problem.

Solution:The completed grid is shown below:The two different ways of filling out the grid are shown below:

A: (red) A square, starting from a box numbered with a square.

B: (green) A cube, starting from a box numbered with a cube.

C: (blue) A prime with an even digit sum, starting from a box numbered with a prime.

D: (yellow) An even number, starting from a box numbered with an even number.