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Have sets of ordered natural numbers represented in square grids been studied? If so, please link me to some papers, articles or other resources about it. For example, properties and transformations in a 5x5 grid. $$\begin{array}{c|c|c|c|c} 1 & 2 &3 & 4 & 5\\ \hline 6 & 7 & 8 & 9 & 10\\ \hline 11 & 12 & 13 & 14 & 15 \\ \hline 16 & 17 & 18 & 19 & 20 \\ \hline 21& 22 & 23 & 24& 25 \ \end{array} $$

Eg, equations for reflections from one element to another.

NOTE: This is not a magic square or a Ulam spiral. Imagine the grid as a coordinate system but instead of say, in a 5x5 grid, (2,4) to get 17, the coordinate would just be (17). I've found equations to get vertical, horizontal, and diagonal reflections for this too; like 17 to 7 (vertical), and 12 to 14 (horizontal). Are there any works directly pertaining to this?

Mathemert
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Some directions you may want to pursue if you want to add material to your project:

  • Compute the determinant of the matrices you have so far (for different sizes: 1,2,3,4,5..). What do you conjecture? Can you prove it?
  • Instead of all natural numbers, put your favorite sequence of natural numbers into a matrix, in the ordered way as above. (Fibonacci numbers, perfect squares, prime numbers, etc.) Do they form patterns when put in the square form? Is there a pattern in the determinants?

If we consider other er numbers, the questions get more difficult and may be better suited for a later project:

For $n > 2$, is there a singular $n \times n$ matrix whose entries are the first $n^2$ prime numbers?

This question was raised here and here.