Magic Squares

This post was inspired by my daily number analysis of my diurnal age. I was struggling to find something of interest about 26102 when I discovered that a $4 \times 4$ magic square could be created in which 26102 was the magic constant. This got me thinking about what numbers could be represented in this way as magic squares.

A $n \times n$ magic square has its $n^2$ cells filled with the numbers from $1 \text{ to } n^2$ in such a way that the sums of all rows, columns and main diagonals are the same. This sum is called the magic constant:

$$\frac{n(n^2+1)}{2}$$ The $3 \times 3$ magic square contains the numbers from 1 to 9 and all rows, columns and main diagonals add up to 15. Thus the magic constant is 15. $$\begin{array}{|c|c|c|} \hline 8 & 1 & 6 \\ \hline 3 & 5 & 7\\ \hline 4 & 9 & 2 \\ \hline \end{array}$$ If we add 1 to each element of the above magic square, we get the next magic square and this has a magic constant of 18: $$\begin{array}{|c|c|c|} \hline 9 & 2 & 7 \\ \hline 4 & 6 & 8\\ \hline 5 & 10 & 3 \\ \hline \end{array}$$ All subsequent $3 \times 3$ magic squares will have magic constants that are multiples of 3. Thus the number 26097 = 3 x 8699 can be represented as: $$\begin{array}{|c|c|c|} \hline 8702 & 8695 & 8700 \\ \hline 8697 & 8699 & 8701\\ \hline 58698 & 8703 & 8696 \\ \hline \end{array}$$ We could represent the various matrices in terms of a subscript for their size and a superscript for their magic sum. This is purely my own invention but it means we would have: M$_3^{15}$, M$_3^{18}$ and M$_3^{26097}$ where M is any matrix that satisfies the condition of being magic and having a particular size and magic sum. The $4 \times 4$ magic square containing the numbers 1 to 15 has a magic sum of 34: $$\begin{array}{|c|c|c|c|} \hline 16 & 2 & 3 & 13 \\ \hline 5 & 11 & 10 & 8\\ \hline 9 & 7 & 6 & 12 \\ \hline 4 & 14 & 15 & 1 \\ \hline \end{array}$$ It would be designated as M$_4^{34}$ in my system. The subsequent magic sums that are possible are 38, 42, 46, 50 and so on. To test whether a number can be the magic constant for a $4 \times 4$ magic square, simply subtract 34 from the number and test whether the result is divisible by 4. In the case of 26102, it is and so we have: $$\begin{array}{|c|c|c|c|}\hline 6533 & 6519 & 6520 & 6530 \\ \hline 6522 & 6528 & 6527 & 6525\\ \hline 6526 & 6524 & 6523 & 6529 \\ \hline 6521 & 6531 & 6532 & 6518 \\ \hline \end{array}$$ This magic square can be designated M$_4^{26102}$. Figures 1 and 2 shows the magic squares together with their magic sums and associated planet or luminary.

Figure 1: source

Figure 2: source

The $5 \times 5$ magic square has a constant of 65 and so any subsequent multiples of 5 will be representable as $5 \times 5$ magic squares. The $6 \times 6$ magic square has a constant of 111 which is 3 x 37. This means that 117, 123, 129 etc. can be represented as $6 \times 6$ magic squares but these numbers are not multiples of 6. To determine whether a number can be the magic constant for a $6 \times 6$ magic square, the 111 must first be subtracted and the result tested for divisibility by 6. This is like the $4 \times 4$ case.

For the $7 \times 7$ magic square, the constant is 175 = 7 x 25 and so any multiple of 7 can be represented by a $7 \times 7$ magic square. For the $8 \times 8$ magic square, the constant is 260 = 4 x 65 which is not divisible by 8 and the situation is as the same as with 4 and 6. For the $9 \times 9$ magic square, the constant is 369 = 9 x 41 so all multiples of 9 above 369 qualify.

Figure 3 shows the magic constants for larger magic squares:

Figure 3: source

So generally, excluding 2, the prime factorisation of a number will quickly tell us what magic squares it can serve as a constant for. For example, 910 = 2 * 5 * 7 * 13 and so it can be represented by magic squares of side 5, 7 and 13. It can also be represented as a magic square with a composite number of sides but a little testing is required. For example, it can be represented as a $4 \times 4$ magic square but not $8 \times 8$. This site is great for such testing.

on May 14th 2021