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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 
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    I don't know what you mean when you write, "the 1 in the top left corner participates in all the $2\times2$ subsquares in specific question."2012-07-25
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    Since I don't understand the condition, I have to ask whether the transpose of your $5\times5$ example is also an example. If it is, then we have proved that it's not true that for $n=5$ there is only one square of this form.2012-07-25
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    So the 1 in the top left corner is involved in a two by two subsquare with every cell in the top row and leftmost column. That is what I meant.2012-07-25
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    Sorry, I still don't understand. In any array of symbols with a 1 in the upper left corner, the 1 in the upper left corner is involved in all those $2\times2$ subsquares. What is different about the involvement, or the subsquares, in the kind of array you want?2012-07-26
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    Nothing. I was only trying to convey that there might be other subsquares that the cells in the top row are involved in as well, that don't contain any of the ones the diagonal.2012-07-26
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    But for us those subsquares are irrelevant.2012-07-26

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