The Magic Squares of Manuel Moschopoulos - The Mathematics of the Methods: Evenly-Even Squares

b. The Methods for Evenly-Even Squares

Moschopoulos' first method for constructing evenly-even squares also works for any square whose side length is a multiple of 4. His description of the placement of dots is unnecessarily complicated, since an easy alternative is to divide the given square into 4×4 subsquares and put dots along the two diagonals of each such subsquare. This is particularly perplexing given that the second method given uses the underlying idea of subdivision into 4×4 subsquares. The rule of placement of numbers given by Moschopoulos is equivalent to writing the numbers from \(1\) to \(n\) from right to left beginning at the right-hand side at the bottom. This is shown for \(n=8\) in Figures 4a and 4b below.

\[\begin{array} {| c | c | c | c | c | c | c | c |} \hline \bullet & & & \bullet & \bullet & & & \bullet \\ \hline & \bullet & \bullet & & & \bullet & \bullet & \\ \hline & \bullet & \bullet & & & \bullet & \bullet & \\ \hline \bullet & & & \bullet & \bullet & & & \bullet \\ \hline \bullet & & & \bullet & \bullet & & & \bullet \\ \hline & \bullet & \bullet & & & \bullet & \bullet & \\ \hline & \bullet & \bullet & & & \bullet & \bullet & \\ \hline \bullet & & & \bullet & \bullet & & & \bullet \\ \hline \end{array}\]

Figure 4a

\[\begin{array} {| c | c | c | c | c | c | c | c |} \hline 64 & 63 & 62 & 61 & 60 & 59 & 58 & 57 \\ \hline 56 & 55 & 54 & 53 & 52 & 51 & 50 & 49 \\ \hline 48 & 47 & 46 & 45 & 44 & 43 & 42 & 41 \\ \hline 40 & 39 & 38 & 37 & 36 & 35 & 34 & 33 \\ \hline 32 & 31 & 30 & 29 & 28 & 27 & 26 & 25 \\ \hline 24 & 23 & 22 & 21 & 20 & 19 & 18 & 17 \\ \hline 16 & 15 & 14 & 13 & 12 & 11 & 10 & 9 \\ \hline 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ \hline \end{array}\]

Figure 4b

\[\begin{array} {| c | c | c | c | c | c | c | c |} \hline 1 & 63 & 62 & 4 & 5 & 59 & 58 & 8 \\ \hline 56 & 10 & 11 & 53 & 52 & 14 & 15 & 49 \\ \hline 48 & 18 & 19 & 45 & 44 & 22 & 23 & 41 \\ \hline 25 & 39 & 38 & 28 & 29 & 35 & 34 & 32 \\ \hline 33 & 31 & 30 & 36 & 37 & 27 & 26 & 40 \\ \hline 24 & 42 & 43 & 21 & 20 & 46 & 47 & 17 \\ \hline 16 & 50 & 51 & 13 & 12 & 54 & 55 & 9 \\ \hline 57 & 7 & 6 & 60 & 61 & 3 & 2 & 64 \\ \hline \end{array}\]

Figure 4c

Now each number \(x\) in a cell containing a dot is replaced by its complement \(n^2+1-x\) as in the example in Figure 4c. Thus, if we choose a column with no dot in the \(i\)th position (from the right) in the bottom row, adding up that column we have \[\left[i+(i+3n)+(i+4n)+(i+7n)+(i+8n)+\cdots +(i+n(n-1))\right]\] \[+\left[\left[(n^2+1)-(i+n)\right]+ \left[(n^2+1)-(i+2n)\right]+ \left[(n^2+1)-(i+5n)\right]\\ + \left[(n^2+1)-(i+6n)\right] + \cdots + \left[(n^2+1)-(i+n(n-2))\right]\right]\] \[=\left[\frac{n}{2}i+n(n-1)+\frac{n}{4}(n^2-n)-n(n-1)\right]\\+\left[ \frac{1}{2}n(n^2+1)-\left[\frac{n}{2}i+\frac{n}{4}(n^2-n)\right]\right],\]

which equals \(\frac{n}{2}\cdot (n^2+1),\) as expected. One can also check the sum of the columns which contain a dot in the \(i\)th position (from the right) in the bottom row. The rows and diagonals are dealt with similarly. It is reasonable to surmise that magic squares of this type were originally built up from the basic 4×4 square using the technique described. From such examples, Moschopoulos' method was derived.

The second method is very poorly described by Moschopoulos and would be impossible to follow without an example. Moreover, he gives no clues as to where or how he obtains the basic 4×4 square which is used in the construction. It is not one that arises from the first method, nor from any row or column permutation of such a square and I have been unable to find it anywhere else. It is however very cleverly constructed since the square given in Fig. 14a (in the translation) contains precisely the numbers from \(1\) to \(32\) thus allowing for the placement of the remaining 32 numbers following the order of the basic square. Each subsquare is also a magic square, and this enables us to see why the method works.