Leonhard Euler, 1707-1783.

Magic squares have been known and studied for many centuries, but there are still surprisingly many unanswered questions about them. In an effort to make progress on these unsolved problems, twelve prizes totalling €8,000 and twelve bottles of champagne have now been offered for the solutions to twelve magic square enigmas.

A magic square consists of whole numbers arranged in a square, so that all rows, all columns and the two diagonals sum to the same number. An example is the following 4×4 magic square, consisting entirely of square numbers, which the mathematician Leonhard Euler sent to Joseph-Louis Lagrange in 1770:

68^{2} |
29^{2} | 41^{2} | 37^{2} |

17^{2} | 31^{2} | 79^{2} |
32^{2} |

59^{2} | 28^{2} | 23^{2} |
61^{2} |

11^{2} | 77^{2} | 8^{2} |
49^{2} |

It's still not known whether a 3×3 magic squares consisting entirely of squares is possible.

The prize money and champagne will be divided between the people who send in first solutions to one of the six main enigmas or the six smaller enigmas listed below. Solutions should be sent to Christian Boyer. His website gives more information about every enigma, and will contain regular updates regarding received progress and prizes won.

Note that the enigmas can be mathematically rewritten as sets of Diophantine equations: for example, a 3×3 magic square is a set of eight equations (corresponding to the three rows, three columns and two diagonals) in ten unknowns (the nine entries and the magic constant to which each line sums).

Here are the six main and six small enigmas:

### How big are the smallest possible magic squares of squares: 3×3 or 4×4?

In 1770 Leonhard Euler was the first to construct 4×4 magic squares of squares, as mentioned above. But nobody has yet succeeded in building a 3×3 magic square of squares or proving that it is impossible. Edouard Lucas worked on the subject in 1876. Then, in 1996, Martin Gardner offered $100 to the first person who could build one. Since this problem — despite its very simple appearance — is incredibly difficult to solve with nine distinct squared integers, here is a question which should be easier:

**Main Enigma 1 (€1000 and 1 bottle):**Construct a 3×3 magic square using seven (or eight, or nine) distinct squared integers different from the only known example and its rotations, symmetries and*k*^{2}multiples. Or prove that it is impossible.

373^{2} | 289^{2} | 565^{2} |

360721 | 425^{2} |
23^{2} |

205^{2} | 527^{2} |
222121 |

### How big are the smallest possible bimagic squares: 5×5 or 6×6?

A *bimagic square* is a magic square which stays magic after squaring its integers. The first known were
constructed by the Frenchman G. Pfeffermann in 1890 (8×8) and 1891 (9×9). It has been proved that 3×3 and 4×4 bimagics
are impossible. The smallest bimagics currently known are 6×6, the first one of
which was built in 2006 by Jaroslaw Wroblewski, a mathematician at Wroclaw University, Poland.

17 | 36 | 55 | 124 | 62 | 114 |

58 | 40 | 129 | 50 | 111 | 20 |

108 | 135 | 34 | 44 | 38 | 49 |

87 | 98 | 92 | 102 | 1 | 28 |

116 | 25 | 86 | 7 | 96 | 78 |

22 | 74 | 12 | 81 | 100 | 119 |

**Main Enigma 2 (€1000 and 1 bottle):**construct a 5×5 bimagic square using distinct positive integers, or prove that it is impossible.

### How big are the smallest possible semi-magic squares of cubes: 3×3 or 4×4?

An *n*×*n semi-magic square* is a square whose *n* rows and *n* columns have the same sum, but whose
diagonals can have any sum. The smallest semi-magic squares of cubes currently known are 4×4
constructed in 2006 by Lee Morgenstern, an American mathematician. We also know 5×5 and 6×6 squares,
then 8×8 and 9×9, but not yet 7×7.

16^{3} | 20^{3} | 18^{3} | 192^{3} |

180^{3} | 81^{3} | 90^{3} | 15^{3} |

108^{3} | 135^{3} | 150^{3} | 9^{3} |

2^{3} | 160^{3} | 144^{3} | 24^{3} |

**Main Enigma 3 (€1000 and 1 bottle):**Construct a 3×3 semi-magic square using positive distinct cubed integers, or prove that it is impossible.**Small Enigma 3a (€100 and 1 bottle):**Construct a 7×7 semi-magic square using positive distinct cubed integers, or prove that it is impossible.

### How big are the smallest possible magic squares of cubes: 4×4, 5×5, 6×6, 7×7 or 8×8?

The first known magic square of cubes was constructed by the Frenchman Gaston Tarry in 1905, thanks to a large 128×128 trimagic square (magic up to the third power). The smallest currently known magic squares of cubes are 8×8 squares constructed in 2008 by Walter Trump, a German teacher of mathematics. We do not know any 4×4, 5×5, 6×6 or 7×7 squares. It has been proved that 3×3 magic squares of cubes are impossible.

11^{3} | 9^{3} | 15^{3} | 61^{3} | 18^{3} |
40^{3} | 27^{3} | 68^{3} |

21^{3} | 34^{3} | 64^{3} | 57^{3} | 32^{3} |
24^{3} | 45^{3} | 14^{3} |

38^{3} | 3^{3} | 58^{3} | 8^{3} | 66^{3} |
2^{3} | 46^{3} | 10^{3} |

63^{3} | 31^{3} | 41^{3} | 30^{3} | 13^{3} |
42^{3} | 39^{3} | 50^{3} |

37^{3} | 51^{3} | 12^{3} | 6^{3} | 54^{3} |
65^{3} | 23^{3} | 19^{3} |

47^{3} | 36^{3} | 43^{3} | 33^{3} | 29^{3} |
59^{3} | 52^{3} | 4^{3} |

55^{3} | 53^{3} | 20^{3} | 49^{3} | 25^{3} |
16^{3} | 5^{3} | 56^{3} |

1^{3} | 62^{3} | 26^{3} | 35^{3} | 48^{3} |
7^{3} | 60^{3} | 22^{3} |

**Main Enigma 4 (€1000 and 1 bottle):**Construct a 4×4 magic square using distinct positive cubed integers, or prove that it is impossible.**Small Enigma 4a (€500 and 1 bottle):**Construct a 5×5 magic square using distinct positive cubed integers, or prove that it is impossible.**Small Enigma 4b (€500 and 1 bottle):**Construct a 6×6 magic square using distinct positive cubed integers, or prove that it is impossible.**Small Enigma 4c (€200 and 1 bottle):**Construct a 7×7 magic square using distinct positive cubed integers, or prove that it is impossible.*(When such a square is constructed, if small enigma 3a of the 7×7 semi-magic is not yet solved, then the person will win both prizes — that is to say a total of €300 and 2 bottles.)*

### How big are the smallest integers allowing the construction of a multiplicative magic cube?

Contrary to all other enigmas which concern the magic squares, this one concerns magic cubes. An
*n×n×n* multiplicative magic cube is a cube whose *n*^{2} rows, *n*^{2} columns, *n*^{2} pillars, and 4 main diagonals
have the same product *P*. Today the best multiplicative magic cubes known are 4×4×4 cubes in which the largest
used number among their 64 integers is equal to 364. We do not know if it is possible to construct a cube
with smaller numbers.

A 4×4×4 multiplicative magic cube by Christian Boyer. Max number=364. *P*=17,297,280.

**Main Enigma 5 (€1000 and 1 bottle):**Construct a multiplicative magic cube in which the distinct positive integers are all strictly lower than 364. The size is free: 3×3×3, 4×4×4, 5×5×5,... . Or prove that it is impossible.

### How big are the smallest possible additive-multiplicative magic squares: 5×5, 6×6, 7×7 or 8×8?

An *n×n* additive-multiplicative magic square is a square in which the *n* rows, *n *columns and two diagonals have
the same sum *S*, and also the same product *P*. The smallest known are 8×8 squares, the first one of which
was constructed in 1955 by Walter Horner, an American teacher of mathematics. We do not know any 5×5,
6×6 or 7×7 squares. It has been proved that 3×3 and 4×4 additive-multiplicative magic squares are impossible.

162 | 207 | 51 | 26 | 133 | 120 | 116 | 25 |

105 | 152 | 100 | 29 | 138 | 243 | 39 | 34 |

92 | 27 | 91 | 136 | 45 | 38 | 150 | 261 |

57 | 30 | 174 | 225 | 108 | 23 | 119 | 104 |

58 | 75 | 171 | 90 | 17 | 52 | 216 | 161 |

13 | 68 | 184 | 189 | 50 | 87 | 135 | 114 |

200 | 203 | 15 | 76 | 117 | 102 | 46 | 81 |

153 | 78 | 54 | 69 | 232 | 175 | 19 | 60 |

**Main Enigma 6 (€1000 and 1 bottle):**Construct a 5×5 additive-multiplicative magic square using distinct positive integers, or prove that it is impossible.**Small Enigma 6a (€500 and 1 bottle):**Construct a 6×6 additive-multiplicative magic square using distinct positive integers, or prove that it is impossible.**Small enigma 6b (€200 and 1 bottle):**Construct a 7×7 additive-multiplicative magic square using distinct positive integers, or prove that it is impossible.

## Comments

## 3 x 3 magic square of squares

Here is an "almost" magic square of squares, where all the rows and columns, and one of the diagonals sum to 21609. Sadly, the diagonal from top left to bottom right sums to 14358

94^2, 2^2 , 113^2

97^2, 74^2, 82^2

58^2, 127^2, 46^2

Arthur Vause

arthur dot vause at gmail.com

## This is just the

This is just the transformation of Lee Sallows's magic square (L. Sal lows, The lost theorem, Math. Intelligencer 19 (1997), no. 4, 51-54).

## 3 x 3 magic square of squares

Here is an "almost" magic square.

All rows and columns, and one diagonal sum to 21609. Sadly the diagonal from top left to bottom right sums to 14358

94^2, 2^2, 113^2

97^2, 74^2, 82^2

58^2, 127^2, 46^2