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:

682 292 412 372 172 312 792 322 592 282 232 612 112 772 82 492 A 4×4 magic square of squares by Euler. An n×n magic square uses n2 distinct integers and has the same sum S for its n rows, its n columns and its two diagonals. Here S=8,515.

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 k2 multiples. Or prove that it is impossible.

3732 2892 5652 360721 4252 232 2052 5272 222121 The only known example of a 3×3 magic square using seven distinct squared integers, by Andrew Bremner. S=541,875.

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 A 6×6 bimagic square by Jaroslaw Wroblewski. S1=408, S2=36,826.

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.

163 203 183 1923 1803 813 903 153 1083 1353 1503 93 23 1603 1443 243 A 4×4 semi-magic square of cubes by Lee Morgenstern. S=7,095,816.

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.

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.

113 93 153 613 183 403 273 683 213 343 643 573 323 243 453 143 383 33 583 83 663 23 463 103 633 313 413 303 133 423 393 503 373 513 123 63 543 653 233 193 473 363 433 333 293 593 523 43 553 533 203 493 253 163 53 563 13 623 263 353 483 73 603 223 An 8×8 magic square of cubes by Walter Trump. S=636,363.

Main Enigma 4 (€1000 and 1 bottle): Construct a 4×4 magic square using distinct positive cubed integers, or prove that it is impossible.

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.

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.

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 n2 rows, n2 columns, n2 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 An 8×8 additive-multiplicative magic square by Walter Horner. S=840, P=2,058,068,231,856,000.