Ed Pegg Jr., November 14, 2005
Sometimes, a puzzle can reach unexpectedly into the past or future. James Nesi had that experience in the late 1950's, when he started putting advertisements into crossword puzzles. The idea was that a potential customer would write down the name of a product, and James did well with this advertising concept. One of his clients was the tourism board of Turkey, so James put in the president of Turkey at the time, Celal Bayar, into the puzzle. A few months later, on May 27, 1960, a coup d'etat ended with the capture of Celal  and prompted the State Department to visit James. "Why are you working for Turkey?", they asked him. James managed to convince the gmen that he was just an innocent puzzle constructor that got paid a small fee to put Turkey into a crossword. The new ruling military committee, meanwhile, sentenced Celal Bayar to death. Before he was killed, a physician examined Celal, then 77, and pronounced him too old and frail to be executed. His sentence was commuted to life imprisonment. Celal lived on, though  released from prison at age 81, and completely pardoned at age 83. He became a hero of Turkey again, outliving all the people that sentenced him to death, and enjoyed celebrity status for the next 20 years, dying in Istanbul at the age of 103 as the most durable head of state in history.
Figure 1. A magic construction of The Great Fubine.
James also showed me the constructions of The Great Fubine. Fubine, real name Cipriano Ferraris, lost everything in the 1929 stock market crash. Quite suicidal, Fubine found solace in huge magic squares and similar designs. I found them interesting as James showed them to me, but I've always been a bit leery of magic squares. I suppose I have no special joy for doing lots of long addition. Isn't that terrible? Fortunately, Clifford Pickover was writing a book, The Zen of Magic Squares, Circles, and Stars, and I let Cliff know about The Great Fubine. Cliff got the whole incredible story from James, and published all of the surviving examples of Fubine's work.
Usually, whenever magic squares come up, I will defer to Harvey Heinz's excellent Magic Squares, Magic Stars & Other Patterns website, or perhaps point to Dürer's Melencholia ("The Relativity of Albrecht Dürer" is a fascinating study). Part of my avoidance was a feeling that magic squares were tapped out. Frenicle de Bessy enumerated the 880 4×4 squares in 1693, Richard Schroeppel enumerated the 275305224 5×5 squares in 1973 (A006052). What else was there?
Figure 2. Detail from Albrecht Dürer's Melencholia
I. Click above for the full image.
Part of my avoidance went away on 17 November 2003. At pretty much the same time, I learned of the 40th Mersenne prime, the Archimedes Stomachion was completely solved, and a 5×5×5 magic cube was discovered. The cube really surprised me, so I started watching for magic square related items that might bear further investigation.
As a followup to my Sudoku column, I looked at various japanesestyle logic puzzles. The excellent Nikoli Cyclopedia of Puzzles demonstrates 215 different logic puzzle varieties. It occurred to me that a multiplicative magic square might make a nice puzzle. Due to the fundamental theorem of arithmetic, multiplying everything out wouldn't be necessary. If the magic product was 5040, for example, or 2^{4}×3^{2}×5×7, a solver could just write the primes that were in each square. Is there a 7 in each column, row, and main diagonal? That's very easy to check. Are there four 2's in each row, column, and main diagonal? Simple. The hardest part would be to make sure that each square contains a different set of primes.
It turns out that many people have looked at multiplicative magic squares. For example, the Stifel paper below is in the library of Benjamin Franklin, who also had an interest in magic squares. People that spoke specifically about multiplicative magic squares include:
The first to be solved, the 3×3 multiplicative magic square, has the property that the magic product K is the cube of the central square. Also, the smallest possible magic product is 216. Rich Schroeppel has a beautiful short proofof both. "Divide the product of the four lines through the center square i by the product of the three across lines. i^{3}=K. Thus, K must be a cube, with at least 9 divisors. 1, 8, 27, 64, 125 have 1, 4, 4, 7, 4 divisors. Done!"
The square below was apparently first found by Sayles, then Dudeney (Stifel and Arnauld built multiplicative magic squares out of powers of 2). I've colored the square by powers of two and three, shown the square as the full factorization into primes, converted it via exponents into ternary numbers, then converted it to base10. Add 1, and you'll get the classic 3×3 addition magic square, the Loshu. The smallest magic constant for a 3×3 multiplicative semimagic square is 120. Rows and columns have a product of 120  the diagonals are ignored. (Puzzle #1: Can you find the 3×3 multiplicative semimagic square with magic constant K=120?)






Will Shortz used the 4×4 multiplicative square for his NPR Puzzle of the Week on October 9, 2005. Over two thousand people sent in solutions. Many people wrote to me specificly to tell me how they solved it  using coins was quite popular. Each row and column has product 5040; or 7!; or 2^{4}×3^{2}×5×7; or 4 pennies, 2 nickels, a dime, and a quarter in each row, column, and main diagonal. As the puzzle is phrased, there are 80 possible solutions.
From Ed Pegg Jr.: In a standard 4 by 4 magic square you arrange the digits from one to sixteen so each row, column and corner digital totals 34. This is a multiplication magic square: Arrange sixteen numbers in a four by four square so that the product of each row, column and corner to corner diagonal is 5,040. You can use any numbers you want. But they have to be whole numbers and you can’t repeat a number in the square. (And as a hint I’ll tell you the number in the upper left corner is 42.) (Puzzle #2)
The best solution for the 4×4 square was found in 1913 by Sayles, also with a magic product of 7!, or 5040. I say better, because the largest number within the square is 28, smaller than the 42 that I used. In 1983 (Discrete Mathematics), Borkovitz and Hwang proved that 5040 is the smallest possible magic product in a 4×4 square. Christian Boyer, Dan Asimov, Michael Kleber, Richard Schroeppel, David Wilson and I didn't initially know of the proof, and tried to find a smaller number that might work. 4320 was the smallest candidate for a magic product, so I spent several minutes moving primes around on grid paper, and found a semimagic 4×4 with product 4320 on the rows and columns (Puzzle #3  find it)  something apparently new.





Sayles published a pandiagonal multiplicative square with constant 14400, pandiagonal meaning that all the broken diagonals give also the same magic product (puzzle #4  find this square). Boyer's computer search proved that the Sayles result was optimal. For more terms: see the 4×4 list referenced in Oct. 2005 under the number A111030 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.
Sayles published the optimal 5thorder pandigital multiplicative square with K=362880=9! (#1), which can be built with two latin squares (producing an Euler square). The smallest possible order5 (#2) magic product is K= 2^{6}3^{3}5^{2}7 = 302400, and was first found by Christian Boyer. He also found the smallest 5×5 semimagic (#3) with K=277200 (and one magic diagonal). Higher products are listed at A111031.



Christian Boyer found the smallest possible magic product for 6×6 multiplicative squares (#1) is K = 25945920 = 2^{6}×3^{4}×5×7×11×13 = 3×13!/6!. This example uses the smallest possible set of integers (1..78) for that magic product. But it is possible to construct 6×6 multiplicative squares with a more compact set of integers (1..66), but a bigger magic product! The second square (#2) has magic product 39916800. Smallest large number in the smallest multiplicative square of order n: 36, 28, 45, 78. This diverges from the sequence for the smallest large number in any ordern multiplicative squares: 36, 28, 45, 66. Further terms in the second sequence are currently unknown.


The latin squares method, used for 4×4 and 5×5 squares, does not work for the 6×6 square due to the famous "36officers problem" of Euler. Because of this, a 6×6 pandiagonal multiplicative magic square may be impossible. Can anyone prove that, or find an example?
Christian Boyer found the 7×7 multiplicative example (#1) with the smallest possible magic product K = 5×6×7×8×9×10×11×12×13×14 = 14!/4! = 3632428800. It uses the smallest set of integers (1..91). Note that the 7 rows are multiplicative magic and additive magic. Magic sum of each row = 218. Unfortunately, the columns and diagonals do not have this property, and it is impossible to get a complete additivemultiplicative 7×7 magic square with K=14!/4!. Based on this, what is the smallest possible additivemultiplicative magic square? The second square (#2) is the smallest known 7×7 pandiagonal multiplicative square, found by Christian Boyer, with magic product 8821612800.


Luke Pebody found that the lowest constant for an 8×8 magic squares is K= 11×12×13×14×15×16×17×18×19×20=20!/10!=670442572800. Simultaneously multiplicative and additive 8×8 and 9×9 squares are known. These are perhaps improvable, with the smallest possibly being 7×7. He also found that the lowest constant for 9×9 magic squares is 10×11×12×13×14×15×16×17×18×19×20×21 = 21!/9! = 140792940288000.


Here is a table of known results, with discoverers (HS = Harry A. Sayles, CB = Christian Boyer, LP = Luke Pebody, EP = Ed Pegg Jr.)
Order  Smallest semimagic constant  Smallest magic constant  Smallest pandiagonal constant  Smallest largest entry 
3  120 (CB)  216 (HS)  Impossible  36 (HS) 
4  4320 (EP)  5040 (HS)  14400 (HS)  28 (HS) 
5  277200 (CB)  302400 (CB)  362880 (HS)  45 (CB) 
6  25945920 (CB)  25945920 (CB)  Impossible?  66 (CB) 
7  3632428800 (CB)  3632428800 (CB)  8821612800? (CB)  91 (CB) 
8  670442572800 (LP)  670442572800 (LP)  ?  160? (LP) 
9  140792940288000 (LP)  140792940288000 (LP)  ?  225? (LP) 
10  43716207959424000? (LP)  43716207959424000? (LP)  ?  290? (LP) 
11  ?  ?  160986670580736000000? (CB)  341? (LP) 
12  ?  ?  ?  444? (LP) 
A seemingly ancient question yielded many new results. Christian Boyer lists various other unsolved questions about magic squares at his site. For example, a magic square is bimagic if it remains magic after squaring each of its integers. The smallest bimagic squares using consecutive integers is known: 8×8. The smallest bimagic squares using distinct integers, not forced to be consecutive, is still unknown! Is it 5×5, 6×6, 7×7? If 5×5 is impossible, is there a proof? Edouard Lucas was the first to work on the subject, in 1891, easily proving that 3×3 is impossible.
Christian Boyer also offers a $100 prize + champagne for the first to solve a simplified version of Martin Gardner's $100 Magic Square of Squares problem. Provide a new example of a 3×3 magic square with 7 distinct square entries, different from the below square and its rotations, symmetries or multiples. Or provide any example with 8 distinct square entries. Here's the only known example in which 7 of the entries are squared integers, found by Andrew Bremner of Arizona State University and independently by Lee Sallows of the University of Nijmegen.
373²  289²  565² 
360721  425²  23² 
205²  527²  222121 
It's quite possible, even likely, that my discovery of the semimagic 4×4 square with constant 4320 has been found before. Many of these discoveries on this topic have been lost to time. If it hadn't been for James Nesi preserving his work, the Great Fubine would likely be entirely forgotten today.
W.S. Andrews, Magic Squares and Cubes, Cosimo Books, p. 283294, 2004.
Christian Boyer, "The Smallest Possible Multiplicative Magic Squares," http://cboyer.club.fr/multimagie//English/Multiplicative.htm.
Suzy Cimino and Jim Sauerberg, "How Many Multiplicative Magic Squares are There?" http://math.stmarysca.edu/Faculty/JS/mult.pdf.
Henry E. Dudeney, Amusements in Mathematics, Dover Books, 1958.
David Finkelstein, "The Relativity of Albrecht Dürer" (pdf), May 2005, http://www.physics.gatech.edu/people/faculty/dfinkelstein.html.
Paul C. Pasles, "More Magic Squares," http://www.pasles.org/magic.html.
Clifford Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton University Press, 2002.
Will Shortz, "NPR Sunday Puzzle, TwiceTold Homophones," http://www.npr.org/templates/story/story.php?storyId=4951886.
Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.
Ed Pegg Jr. is the webmaster for mathpuzzle.com. He works at Wolfram Research, Inc. as an associate editor of MathWorld, and as administrator of the Mathematica Information Center.