# A True Diagonal Problem

How many ways are there to place the numbers 1, 2, 3, 4, 5 in the 16 squares of a 4×4 grid so that each row, each column, and each diagonal consists of distinct numbers?

Here, two placements are considered different if they are different in any way. And diagonal means true geometric diagonal: any sequence of squares in the NW-SE or NE-SW directions. There are 10 such diagonal lines.

An equivalent statement is to assume one has several chess queens, in five different colors. A legal placement would be a placement of 16 queens on the 4×4 board so that no queen attacks another queen of the same color.

Comment: An old problem (#308) in Dudeney's famous Amusements in Mathematics book asks for the placement that maximizes the sum of the placed numbers. Here we are asking for all legal placements.

Source: Very old problem of Joe Konhauser when he worked at Ford Motor Co. Joe was the founder (1968) of our PoW program.

© Copyright 2010 Stan Wagon. Reproduced with permission.

13 September 2010