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

That part concluded, it is now time to speak about the placement (of the numbers). Let us commence with the simplest case. One can form first of all a square based on the number 3 (Fig. 2) and we will speak firstly about this, however the method about to be expounded in regard to this square, will apply to all squares of similar type¹⁴ . One can make an arrangement giving (the sum) equal in each direction, using [the method of] two's and three's and it is possible also to use [the method of] three's and five's.

The  Method  of  Two's and  Three's

The [method of] two's and three's is as follows:

Suppose the cells of the smallest possible square are drawn up, that is to say, of the (square corresponding to the) number 9, thus. We place the number one in the middle cell of the three along the bottom row, and we count two cells, including that which holds the number one and then the next below this in a straight line. The general rule is always to go to the lower one. Since however, we do not find (such a cell), we return again to the top (of the square) in a straight line moving through the cells in a cyclic fashion. We count that cell as the second one. Then we place 2 in the cell to the right of this in a straight line.

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

Figure 2

We again count two cells, the one which holds the 2 and another below it and we look to the cell on the right in a straight line so that we might place the number 3; the general rule is always to move to the right. Since we do not find (such a cell), we return to the left (hand side of the square) in a straight line. For whenever a row of the square ends you return to the beginning (of that row)¹⁵ . We place 3 in the last cell as we cycle back, that is, in the first cell as we move to the right, the one which we come to as we count the cells from the beginning in circular movement. Now since we have come to the number 3, which when multiplied by itself gives the (area of the) square, that is to say, it is the side of the number 9, we no longer count two places in order to put 4 in the cell on the right, but three places, as follows. We count the first as that which holds 3, the cell below this as the second as we take as third the cell directly below, and since we do not find (a cell there), we return to the top moving in a straight line, and count that one as the third. In that very cell we place 4, not turning aside (to the right). Following from this and taking that cell as our starting point we count again two cells and place the next number in turn (in the cell) on the right, following the previously explained pattern, and we continue with this until we again come to [a multiple of] the side of 9, that is to say, to 6, which is twice 3. Again, having come to this number, we count three cells and place the next number in turn in the third cell, not turning aside (to the right). Once again we count two and we make the placement in the cell on the right. We continue doing this to the end. As before, we count through two cells for all the other numbers, but through three cells precisely when we come upon a multiple of the side¹⁶. This is the procedure for all squares of the same form, for we count according to this pattern; by two's until we reach the cells containing (a multiple of) the side of the given square, and by three's whenever we come upon these numbers in turn, and we continue thus till the end, cycling back through the cells, as above.

All the squares (of this type) simply follow the same pattern except in regard to the placing of the number one, for this is not always put in the same place but assumes a different position in each square. In the first square which can be constructed from an odd number, it is placed in the middle of the cells along the bottom. In the second such square, (it is placed) in the middle cell of the row one up from the bottom, and in the third, in the middle of the row one further up. Clearly, as the numbers increase, so the number one rises up through the cells. It always transpires that it is placed in the cell which lies directly below the cell in the middle of the given square of that type. One can see all these things more clearly in the diagram. (Figs. 3,4,5).

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

Figure 3

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

Figure 4

\[\begin{array} {| c | c | c | c | c | c | c |} \hline 37 & 78 & 29 & 70 & 21 & 62 & 13 & 54 & 5 \\ \hline 6 & 38 & 79 & 30 & 71 & 22 & 63 & 14 & 46 \\ \hline 47 & 7 & 39 & 80 & 31 & 72 & 23 & 55 & 15 \\ \hline 16 & 48 & 8 & 40 & 81 & 32 & 64 & 24 & 56 \\ \hline 57 & 17 & 49 & 9 & 41 & 73 & 33 & 65 & 25 \\ \hline 26 & 58 & 18 & 50 & 1 & 42 & 74 & 34 & 66 \\ \hline 67 & 27 & 59 & 10 & 51 & 2 & 43 & 75 & 35 \\ \hline 36 & 68 & 19 & 60 & 11 & 52 & 3 & 44 & 76 \\ \hline 77 & 28 & 69 & 20 & 61 & 12 & 53 & 4 & 45 \\ \hline \end{array}\quad{Side}=369\]

Figure 5