The Magic Squares of Manuel Moschopoulos - The Method for Odd Squares - Three's and Five's

(The method of counting by) three's and five's is as follows: We draw the square and we draw in it the cells of the number square. We then place the number one always in the middle of the topmost (row of) cells. We count three cells, including the one which holds the number one, and two below this in turn, and we place 2 in the cell directly on the right of the third cell. We again count from there three cells in a similar fashion and in the cell to the right we place 3. If there is no cell on the right, we return to the left side in a straight line, as in the previous method, and we place it in the last cell as we cycle back, that is, in the first cell as we move to the right. We continue to do this until we come to the side of the given square, and, after we have reached this number, we count five cells, including the cell having the side length and four below it. Then we place in the fifth cell the next number in turn after the side length, not turning (to the right), and again we count by three's until we reach (a multiple of) the side, cycling through the cells as in the previous method. We continue this to the end. This method is in all respects similar to the previous one except that there, the number one was placed in a different position for each different square, but here it is always (placed) in the middle of the top (row of) cells. There, moreover, we counted by two's and three's, while here by three's and five's. One can see these things in the diagram. (Figs. 6, 7, 8)

\[\begin{array} {| c | c | c |} \hline 8 & 1 & 6 \\ \hline 3 & 5 & 7 \\ \hline 4 & 9 & 2 \\ \hline \end{array}\quad{Side}=15\]

Figure 6

\[\begin{array} {| c | c | c | c | c |} \hline 10 & 18 & 1 & 14 & 22 \\ \hline 4 & 12 & 25 & 8 & 16 \\ \hline 23 & 6 & 19 & 2 & 15 \\ \hline 17 & 5 & 13 & 21 & 9 \\ \hline 11 & 24 & 7 & 20 & 3 \\ \hline \end{array}\quad{Side}=65\]

Figure 7

\[\begin{array} {| c | c | c | c | c | c | c |} \hline 38 & 14 & 32 & 1 & 26 & 44 & 20 \\ \hline 5 & 23 & 48 & 17 & 42 & 11 & 29 \\ \hline 21 & 39 & 8 & 33 & 2 & 27 & 45 \\ \hline 30 & 6 & 24 & 49 & 18 & 36 & 12 \\ \hline 46 & 15 & 40 & 9 & 34 & 3 & 28 \\ \hline 13 & 31 & 7 & 25 & 43 & 19 & 37 \\ \hline 22 & 47 & 16 & 41 & 10 & 35 & 4 \\ \hline \end{array}\quad{Side}=175\]

Figure 8

These are the methods in regard to squares formed from odd numbers.