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Looking to trade magic cards for money? Look no further! Whether you're a seasoned collector or a casual player, there are plenty of options available to turn your magic cards into cash. With the growing popularity of online marketplaces and local gaming communities, finding buyers for your cards has become easier than ever. **The main idea here is that there are opportunities nearby to exchange magic cards for money**. One of the most popular platforms for buying and selling magic cards is online marketplaces such as eBay, TCGplayer, and Cardmarket. These platforms offer a wide range of options for both buyers and sellers and provide a secure platform to complete transactions. **The idea here is that online marketplaces offer a convenient and secure way to trade magic cards for money**.

Plug in another progression:
9409, 12769, 16129

Another approach is Lee Morgenstern s way to create 3x3 magic squares from arithmetic progressions with different distances, but have a perfect square in common. col row 2, 58 - 3360 3364 - 4 58 58 - 2 2 11, 59 - 3360 3481 - 121 59 59 - 11 11 19, 61 - 3360 3721 - 361 61 61 - 19 19 22, 62 - 3360 3844 - 484 62 62 - 22 22 41, 71 - 3360 5041 - 1681 71 71 - 41 41 46, 74 - 3360 5746 - 2116 74 74 - 46 46 58, 82 - 3360 6724 - 3364 82 82 - 58 58 74, 94 - 3360 8836 - 5475 94 94 - 74 74 97, 113 - 3360 12769 - 9409 113 113 - 97 97 113, 127 - 3360 16129 - 12769 127 127 - 113 113 134, 146 - 3360 21316 - 17956 146 146 - 134 134 163, 173 - 3360 29929 - 26569 173 173 - 163 163 206, 214 - 3360 45796 - 42436 214 214 - 206 206 277, 283 - 3360 80089 - 76729 283 283 - 277 277 418, 422 - 3360 178084 - 174724 422 422 - 418 418 839, 841 - 3360 707281 - 703921 841 841 - 839 839.

**The idea here is that online marketplaces offer a convenient and secure way to trade magic cards for money**. If you prefer face-to-face transactions, local gaming stores and gaming clubs can be excellent places to find buyers for your magic cards. Many stores and clubs host trading events or tournaments where you can showcase your collection and connect with potential buyers.

Win money with magic squares

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:

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.

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.

• 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?

• 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.

• 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.

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