Ken's Puzzle of the Week

Latin and Euler Squares with Diagonals

1. Into a 5x5 grid, place the numbers 1 to 5 once in each row, column, and main diagonal, creating a Latin Square (with additional diagonal restrictions.)  Disregarding rotations, reflections and substitutions, how many unique solutions are there?  Let the top row be "1 2 3 4 5".

2. Add to the above grid the letters A to E once in each row, column and main diagonal.  Also, all grid entries are unique, creating an Euler Square (with additional diagonal restrictions).  How many solutions are there?  Let the top row be "1A 2B 3C 4D 5E".

Extension: Repeat for 3x3, 4x4, 6x6, and 7x7.
Extension 2: How many Euler Squares are there of each size, without the diagonal restriction?  (I couldn't find a good list, so perhaps someone else knows of a site that displays them?)

Source: Original.  When checking uniqueness, remember to consider "What if all 1s and 5s changed to 5s and 1s?"  There are fewer solutions than you might expect.
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