A proper Euler magic matrix of order 6

In the vast landscape of mathematics, there is a particular fascination with grids of numbers that obey strict, hidden rules. Imagine a square table filled with whole numbers. If you add up the numbers in any row, any column, or along either of the two main diagonals, the total is always the same. This is a magic square, a puzzle that has intrigued thinkers for centuries. But mathematicians often push further, asking what happens if the numbers inside the square are not just any integers, but perfect squares themselves—numbers like one, four, nine, or sixteen. Even more challenging is a specific type of grid where the rows and columns are not just balanced in sum, but are also mathematically independent of one another, meaning they do not overlap in a way that creates redundancy. This combination of requirements creates a rare and difficult object known as an Euler magic matrix. For a long time, mathematicians knew these grids existed for certain sizes, such as four by four or eight by eight, but a specific size in the middle remained a mystery. The question was simple yet stubborn: could such a grid be built using six rows and six columns?

A researcher named Sanjit Singh Mehat has now answered that question with a definitive yes. In a recent study, Mehat constructed the first known example of a proper Euler magic matrix of order six. To understand the significance, one must look at the strict conditions required. The grid must contain thirty-six whole numbers. When these numbers are squared, the sum of the six numbers in every single row must equal a specific total. The same total must appear in every column. Furthermore, the sum of the six squared numbers running from the top-left corner to the bottom-right corner must match that total, and the sum of the six squared numbers running from the top-right to the bottom-left must also match. Finally, and perhaps most importantly, the grid is considered "proper" only if the absolute values of all thirty-six numbers are completely different from one another. No two numbers can have the same size, even if one is positive and the other is negative.

For decades, the existence of such a grid for a six-by-six layout was unknown. Previous work had settled the cases for smaller grids, proving that a three-by-three version was impossible, and confirming that versions for sizes one, two, four, five, and eight did exist. The six-by-six case stood as the smallest missing piece in the puzzle. Mehat's work fills this gap by presenting two distinct, concrete examples of these grids. The first example uses a specific total sum of 18,500 for the squared entries. The second example, found independently, uses a different total sum of 43,290. Both matrices are filled with integers that, when squared and added up according to the rules, yield these exact totals. The researcher verified that in both cases, the rows and columns are mathematically independent, the diagonal sums are correct, and every single number in the grid has a unique magnitude.

The path to finding these grids was not a matter of simple trial and error. The search space is so vast that checking every possibility by hand or with standard computing methods would be impossible. Mehat developed a specialized method to narrow the search. Instead of trying to build the entire grid at once, the approach involved generating smaller building blocks and combining them in ways that satisfied the row and column rules first. Once a candidate grid was found that met those basic requirements, the researcher then looked for a specific arrangement of the rows and columns that would also satisfy the diagonal rules. This strategy proved effective, allowing the discovery of the two examples within minutes of running the search program. The study also established a mathematical lower limit for the total sum in such a grid, proving that the sum of the squared entries must be at least 2,485, a boundary that helped guide the search.

To ensure the results were beyond doubt, the findings were subjected to rigorous verification. The calculations were checked using exact integer arithmetic, a method that leaves no room for rounding errors. A separate, independently written program confirmed the results, and the entire proof was also verified by a computer system designed to check mathematical logic. This triple-check process confirms that the grids are real and that they meet every condition required. The work does not rely on complex theories or unproven assumptions; it rests on the explicit display of the numbers themselves. Anyone with a calculator can verify that the rows, columns, and diagonals of the provided grids add up correctly and that no two numbers share the same size.

This discovery resolves a specific, long-standing question in the field of combinatorial mathematics. While the existence of magic squares made of squares was already known for six-by-six grids, those earlier results did not guarantee the strict independence of rows and columns required for an Euler magic matrix. Mehat's work demonstrates that such a structure is indeed possible, adding the number six to the list of sizes where these special grids can be built. The study provides the actual numbers for two such grids, offering a concrete solution to a problem that had remained open. It stands as a clear example of how modern computational methods, when guided by clever mathematical strategies, can solve problems that have eluded researchers for years, turning a theoretical possibility into a tangible reality.