Rules: Fillomino variant. Instead of grouping cells with their orthogonal neighbors, cells are grouped with their diagonal neighbors. Two cells belong in the same group if they touch diagonally (that is, share a corner but not an edge) and have the same number. A group must have as many cells as the number its cells have. (Notice that this means two groups with the same number can’t touch diagonally, just as two groups with the same number can’t share an edge in regular Fillomino.) Additionally, cells with the same number can’t share an edge.
Equivalently, the white cells must form a valid Fillomino when the grid is rotated 45 degrees, as must the lightly shaded cells, and two cells that share an edge must contain different numbers.
The chequerboard shading is purely aesthetical.
As far as I know, this variant is my own invention.