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?

(In reply to Solution by Federico Kereki)

...Also in reply to Penny's solution

There are not 2304 4x4 Greco-latin squares.

Among the combinations you include, this would be one:

A1 B2 C3 D4
C3 D4 A1 B2
D4 C3 B2 A1
B2 A1 D4 C3

A brief look at this shows that it does not count.

Edit:  Interestingly enough, looking over the problem, it does not directly preclude this combination.  It was more clear before Victor edited it in the queue that the 16 squares should be A1, A2, A3, A4, B1, etc.

I say that there are actually half that many squares, 1152!

The reason is that in your calculations, there are two configurations.  If the letters follow the first configuration, then the numbers follow the second, and vise versa.  Only half of the 2304 combinations follow this condition, so only half are valid.

 

Edited on May 3, 2004, 12:56 am

Posted by Tristan on 2004-05-02 23:55:44