Latin squares

A latin square is composed of n² squares containing n symbols (latin letters) with every symbol once and only once in every line and collumn.

For instance :

 a  b  c  d 
 d  c  b  a 
 b  a  d  c 
 c  d  a  b 
There are other 4×4 latin squares. Two latin squares [aij] and [bij] are said to be orthogonal if we can merge them into one square [aijbij], every composite symbol (a,b) being only once.
By tradition, one is made of latin letters and the other of greek letters, to build a graeco-latin square.

    as the well known 10×10 graeco-latin square in the picture, said to be impossible by Euler, and only discovered in 1960 ! 

Merge three 4×4 latin squares to build an "arabo-graeco-latin" square, build of combinations of arabic numbers 1234, greek letters αβγδ and latin letters abcd, every symbol once and only once in every row and column, and every symbol pair only once.       Solution      Details

 

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