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?

There are 2,304 unique Greco-Latin squares.

Explanation:

There are 4 numbers and 4 letters to consider. If we just worry about the 4 numbers, and then square that result, we should have the final answer.

So the numbers are 1,2,3,4.

There are 4*3*2*1=24 possibilities for the main diagonal.

From the symmetry of these numbers, if we can analyze any one of these diagonals, we should have an answer.

Consider main diagonal 1-2-3-4.

Then the only possibilities for the four rows are:

{1423...3241...4132....2314} and {1342...4213...2431....3124}

So every diagonal possibility has 2 and only 2 configurations.

There are then 48 unique squares with respect to the 4 numbers. By squaring that figure to add the four letters, we get all the unique squares:

48*48=2304     

Posted by Penny on 2004-05-02 19:21:50