Here are two $$5 \times 5$$ Latin-Square Adjacent puzzles. For each cell, put a digit between $$1$$ and $$5$$ such that:

• Each row and column contains one occurrence of each digit $$1$$ to $$5$$.
• If two cells are adjacent and their digits are numerically adjacent (i.e. one higher or one lower), then there will be a (black) border between them.
• All borders are given, thus two cells that are adjacent and not having a border between them means their digits are not numerically adjacent.

In the second row, the number $$2$$ can only go in the third or fifth cell which deduces the fourth cell as $$5$$ and the second cell in the first row follows as $$5$$ as well.
Now the first row can only be filled by $$2,5,3,1,4$$ and the second row follows with $$1,4,2,5,3$$. The rest of the grid can then be filled accordingly.