Consider a Latin square puzzle played not necessarily in a $10\times 10$ grid, but in some $n\times n$ grid of the following form:
1 2 3 ... n 2 1 3 1 . . . . . . n 1
such that all cells in the first row are participating in a "subsquare"- ie an arrangement where the digits in $(p,q)$ and $(a,b)$ contain the same digit, and $(p,b) and (a,q)$ contain the same digit. In this construction the $1$ in the top left corner participates in all the $2 \times 2$ subsquares in specific question. For $n=4$, here is one such arrangement:
1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1
For $n=5$ here is one such square:
1 2 3 4 5 2 1 4 5 3 3 5 1 2 4 4 3 5 1 2 5 4 2 3 1
For $n=6$ here is one such arrangement:
1 2 3 4 5 6 2 1 4 3 6 5 3 5 1 6 4 2 4 6 5 1 2 3 5 3 6 2 1 4 6 4 2 5 3 1
For $n=7$ here is one such arrangement:
1 2 3 4 5 6 7 2 1 4 5 6 7 3 3 7 1 2 4 5 6 4 6 7 1 2 3 5 5 3 6 7 1 2 4 6 4 5 3 7 1 2 7 5 2 6 3 4 1
For $n=8$ here is one such arrangement:
1 2 3 4 5 6 7 8 2 1 4 3 6 5 8 7 3 7 1 8 4 2 5 6 4 8 7 1 2 3 6 5 5 6 8 2 1 7 3 4 6 5 2 7 8 1 4 3 7 2 5 6 3 8 1 2 8 4 6 5 7 4 2 1
For $n=9$ here is one such arrangement:
1 2 3 4 5 6 7 8 9 2 1 4 8 6 5 3 9 7 3 7 1 2 4 8 9 5 6 4 6 7 1 9 2 8 3 5 5 9 8 6 1 3 2 7 1 6 4 9 3 7 1 5 2 8 7 8 5 9 2 4 1 6 3 8 5 6 7 3 9 4 1 2 9 3 2 5 8 7 6 4 1