The gouda in E3 tells us it’s adjacent to all 5 types of cheese. Since brie is never adjacent to feta, feta must be in E2. This also means D3 and E4 can be either gouda or stilton.

The feta in E2 tells us it’s adjacent to 2 of its own kind. Since there cannot be any feta in D3, this must mean D1 and E1 are feta.

The mozzarella in D2 tells us it’s adjacent to all 5 kinds of cheese. Since brie is never adjacent to feta, there’s only one place it can go: C3.

The mozzarella in D2 should also be adjacent to one other mozzarella. This cannot go to C2 (row 2 only contains 1 of each cheese) or D3 (E3 is adjacent to all 5 kinds of cheese) so it goes to C1.

The feta in D1 tells us row 2 is the only row to contain all 5 kinds of cheese. C2 therefore can be a brie, a gouda or a stitlon. However, since A3 is a stilton, and stiltons are never adjacent to another stilton, C2 must be a stilton.

Since stiltons are never adjacent to another stilton, we now know that D3 is gouda and E4 is stilton.

The stilton in C2 tells us it’s adjacent to 3 mozzarellas. Since mozzarellas are adjacent to max 1 of their own kind, the third mozzarella must be in B3.

We can now place the final feta – it cannot be adjacent to any bries, and we know either A2 or B2 is a brie. The only possible places are A4 and A5. Since there’s already a feta in the corner E1, the last feta goes to A4.

B4 cannot be a feta, cannot be a brie, cannot be a stilton and cannot be a mozzarella, so it must be a gouda.

C4 cannot be a feta, cannot be a mozzarella, cannot be a gouda, cannot be a brie, so it must be a stilton.

The gouda in B4 tells us it’s adjacent to one other gouda. This can only be C5, as there are no goudas in the corners.

The feta in B5 tells us it’s adjacent to 4 kinds of cheese. Since it cannot be adjacent to a brie, A5 must be a mozzarella.

We can now place the final stilton, which cannot be adjacent to any other stiltons. The only possible place is A1.

We can now place the final mozzarella, which cannot be in A2 or A3, nor in E5, so it must go to D5.

The brie cannot go to B2, as there would be no possible option for B1, so it must go to B1 and A2. B2 is a gouda.

The final corner is a brie.