In Newtown Middle School, there was a school boy named Chris who was an absolute troublemaker. One of his many schemes against the school was drawing squares in permanent marker on all the walls in the school. Thus, when the school decided to make a punishment for Chris, they decided to do something involving squares.

The school made Chris create all possible unique Greco-Latin squares using A-D and 1-4. (A 4x4 Greco-Latin square using A-D and 1-4 is a special 4x4 square. Each cell of the square has exactly one letter of the four and one number of the four within it. The end result will have every letter and every number used once in each row, column, and main diagonal of the square.)

Chris is a very slow boy, and after several hours, he figured out all the possible Greco-Latin squares. How many squares did he find?

Working first with letters, I found only two possible configurations:

A B C D A B C D

C D A B D C B A

D C B A B A D C

B A D C C D A B

so we get 48 (=4!+4!) possible arrangements for the letters. Likewise, we'll have other 48 arrangements for the numbers. So the total number is 48²=2304.