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:

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.

682	292	412	372
172	312	792	322
592	282	232	612
112	772	82	492

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.

The only known example of a 3×3 magic square using seven distinct squared integers, by Andrew Bremner. S=541,875.

3732	2892	5652
360721	4252	232
2052	5272	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.

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

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.

A 4×4 semi-magic square of cubes by Lee Morgenstern. S=7,095,816.

163	203	183	1923
1803	813	903	153
1083	1353	1503	93
23	1603	1443	243

• 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.

An 8×8 magic square of cubes by Walter Trump. S=636,363.

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

• 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 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.

• 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.

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

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.