Logic Masters Deutschland e.V.

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English:
1. Row Rule: Each row is a permutation of the vector C1=(1,2,…8,9).

​ 2. Column Rule: Each column is a permutation of the transpose of the vector C2=(1,2,…8,9).

​ 3. Submodules: The entire 9×9 grid is partitioned into 9 non-overlapping 3×3 submatrices. Among these, 8 are invertible matrices, and 1 is a singular (non-invertible) matrix. Each submodule contains distinct numbers with no repetition.

​ 4. Rank Constraint: The rank of the singular 3×3 submatrix is equal to the number that should be placed in the cell marked with "?". Furthermore, this number must not appear in the same relative position across all 9 submodules. (For example, if the rank of the singular matrix is n and the top-left cell of a submodule is n, then the top-left cell of every other submodule cannot be n.)

​ 5. Inversion Count Rule: The numbers outside the grid indicate the number of inversions in their corresponding row or column vector.

​ 6. Core Matrix Constraint: There exists a 3×3 matrix A=(aij) that is orthogonally adjacent (horizontally or vertically) to the singular matrix from Rule 4, satisfying:

- The determinant of A, det(A), equals a33.
​ - The tens digit of the determinant of the adjugate matrix det(A*) is a32, and the units digit is a23.
​ - Let B=(bij) be the square matrix formed by the minor of a33 in det(A). Then the product of det(B^(-1)) (the determinant of the inverse of B) with b12 and b22 must both be integers.

7. Matrix Transpose and Submodule Correspondence: Each element (A^T)_{i,j} in the transpose of matrix A (which is aji in the original matrix A) must equal the element in the corresponding position of all submodules from Rule 3.

- Example: The element in the 3rd row, 2nd column of the 3rd submodule from the top and 2nd from the left equals a23 in matrix A (i.e., (A^T)_{3,2}).

Chinese:
1.行规则：每行数字为向量C1=(1,2,…8,9)的一个置换。

2.列规则：每列数字为向量C2=(1,2,…8,9)的转置的一个置换。

3.子模块：整个9×9网格可划分为9个互不重叠的3阶矩阵，其中8个为可逆矩阵，1个为不可逆矩阵；每个矩阵内无重复元素。

4.秩约束：该不可逆3阶矩阵的秩，等于“?”位置应填的数字；且该数字在所有9个子模块的相同相对位置上均不重复（例如，若不可逆矩阵的秩为n，且已知R1C1为n，则每个子模块的第1行第1列元素都不能是n）。

5.逆序数规则：网格外的数字，对应其所在行或列向量的逆序数。

6.核心矩阵约束：存在一个与规则4的不可逆矩阵正交相邻的3阶矩阵A=(aij) ，满足：
- 矩阵的行列式det(A) = a33；
- 其伴随矩阵的行列式det(A*)值的十位数为a32，个位数为a23；
- 记其行列式的a33的余子式构成的方阵为B=(bij) ，则B的逆矩阵的行列式det(B^(-1))与b12、b22的乘积均为整数。

7.矩阵转置与宫格对应：规则6中矩阵A的转置A^T中每个元素(A^T)_{i,j} （即原矩阵A的元aji），与规则3中所有子模块的对应位置元素相同。
- 示例：从上往下数第3个、从左往右数第2个子模块的第3行第2列元素，等于A的元素a23（即(A^T)_{3,2}）。