Four queens on a four-by-four board

Question

A queen attacks along her row, her column and both diagonals. 1. In how many ways can four queens be placed on a 4 by 4 board so that no two attack each other? Find them by hand. 2. Why can no queen be in a corner? How many solutions does the 8 by 8 board have?

Accepted Answer

Two arrangements, mirror images of each other. The 8 by 8 board has 92. Setting up. With four queens and four rows, each row holds exactly one queen, and likewise each column. So a solution is a placement of one queen per row in distinct columns, with no two on a diagonal. Row 1, column 1. A queen at (1, 1) attacks the whole main diagonal, so the queen in row 2 must be in column 3 or 4. Column 3: row 3 could then only use column 2 or 4 (columns 1 and 3 are taken), and both are attacked diagonally by the queen at (2, 3). Dead end. Column 4: (2, 4) leaves row 3 with column 2 (column 3 is attack…

Four queens on a four-by-four board

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