The 2-(64,8,1) Design

The combinatorial design 2-(64,8,1) can be represented with four squares. For any 2 integers chosen up to 64, there is exactly 1 length-8 row, column, or diagonal that contains the pair. Each square can be modified by one of the 8!! = 384 permutations that preserve rows, columns, and diagonals. Combinatorially, 2-(64,8,1) is unique.

The 2-(16,4,1) design can be found in the Magic Squares and Designs Demonstration, and is also combinatorially unique. Designs exist for 2-(25,5,1), 2-(49,7,1), 2-(81,9,1), 2-(121,11,1), and 2-(169,13,1), but they would not be squares with diagonals. No designs exist for either 2-(36,6,1) or 2-(100,10,1). Whether 2-(144,12,1) exists is a long-unsolved question.