Magic squares of powers
A magic square of order n is an n by n grid of integers in which each row, each column, and each of the two main diagonals adds up to the same number. The picture above shows a magic square of order 6 that was discovered by Jaroslaw Wroblewski in 2006. The grid has the property that each row, column, and main diagonal adds up to the same number: 408 in this case. The remarkable thing about it is that if we raise each of the entries to the power 2, we obtain another magic square in which each row, column, and main diagonal adds up to 36826.
A magic square whose squared entries give rise to another magic square is called a bimagic square. It is easy to form trivial examples of bimagic squares, for example by setting all the entries equal to 1. Another way to form trivial examples is to use a doubly diagonal Latin square, like the example in the above picture. A Latin square, like a completed Sudoku puzzle, is an n by n grid is filled with n different types of symbols in such a way that each row and each column contains each symbol exactly once. A doubly diagonal Latin square satisfies the additional property that each of the two long diagonals contains each symbol exactly once. In order to avoid uninteresting solutions like this, we say that a bimagic square of order n is nontrivial if it uses strictly more than n different symbols.
Another variant of this problem is to try to construct an n by n bimagic square using the consecutive integers 1, 2, 3, up to n2. The picture above shows a magic square of order 12 whose entries are the consecutive integers 1 up to 144, which was discovered by Walter Trump (possibly no relation) in 2002. Trump’s example is not only magic, but bimagic and trimagic: not only does squaring all the entries give another magic square, but cubing all the entries gives another magic square.
More generally, a K-multimagic square is a magic square whose k-th powers are magic squares for all k=2, 3, up to K; for example, Wroblewski’s example is a 2-multimagic square, and Trump’s example is a 3-multimagic square. The recent paper A circle method approach to K-multimagic squares by Daniel Flores makes some significant advances in this area. Flores defines the integer N2(K) to be the smallest number N for which there exists a nontrivial multimagic square of order N. The previously known best upper bounds for N, which are shown in the table above, include the known results that Wroblewski’s order 6 example is the smallest possible size of a 2-multimagic square, and Trump’s order 12 example is the smallest possible size of a 3-multimagic square.
Flores proves that in general, N2(K) is less than or equal to 2K(K+1)+1. The previously known bound was exponential in K, and Flores’ bound is far better than the previously best known results even for K=4, 5, and 6. Despite the recreational appearance of the problem, Flores’ proof uses some sophisticated techniques from analytic number theory, such as the Hardy–Littlewood circle method.
Another recent paper on a related problem is On the existence of magic squares of powers by Nick Rome and Shuntaro Yamagishi (with an appendix by Diyuan Wu). The authors posted their paper in draft form after seeing Flores’ paper the previous day. Rome and Yamagishi are interested in the problem of constructing a magic square whose entries are squares. The picture above is a famous example of such a square that was constructed by Leonhard Euler in 1770. (Note that this example does not, and is not required to produce a magic square if the powers of 2 are removed.)
It was a conjecture by Anthony Várilly-Alvarado that for every n at least 4, there exists a magic square of order n whose entries are square numbers. Rome and Yamagishi prove this conjecture, again by using the Hardy–Littlewood circle method (named after G.H. Hardy and J.E. Littlewood, pictured above). Rome and Yamagishi also prove similar results for magic squares of cubes, fourth powers, and so on, for sufficiently large values of the order n, although they do not give a precise bound for what “sufficiently large” means.
It is not known whether there exists a 3 by 3 magic square whose entries are squares. This existence problem was originally posed in 1876, but was popularized in 1996 by Martin Gardner, who offered a cash prize for the first person to construct such a square. The problem remains open to this day, but it is known that if a solution exists, the numbers involved must be huge.
Picture credits and relevant links
The website multimagie.com by Christian Boyer contains a wealth of information on multimagic squares.
The order 12 magic square by Walter Trump comes from his website.
The order 4 magic square of squares comes from the paper by Nick Rome and Shuntaro Yamagishi.
The table of K-multimagic square records and the statement of Theorem 1.2 come from the paper by Daniel Flores.
The graphics of the two order 6 magic squares are my own work.
I do not know where the photograph of Hardy and Littlewood originally came from.
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