This took me rather a long time to put together, so I'd better post it so somebody else can use it. This is the arrangement for a 10x10 Graeco-Latin square. The one I'm in the middle of putting together uses outer squares of 10 fabrics the right way round (the capital letters) with an inner square of the same 10 fabrics reversed (the lower-case letters).
Bb Ih Jg Af Ca Ej Gi Dc Fd He Hi Cc Ib Jh Ag Da Fj Ed Ge Bf Gj Bi Dd Ic Jb Ah Ea Fe Hf Cg Fa Hj Ci Ee Id Jc Ab Gf Bg Dh Ac Ga Bj Di Ff Ie Jd Hg Ch Eb Je Ad Ha Cj Ei Gg If Bh Db Fc Ig Jf Ae Ba Dj Fi Hh Cb Ec Gd Cd De Ef Fg Gh Hb Bc Ii Ja Aj Df Eg Fh Gb Hc Bd Ce Aa Ij Ji Eh Fb Gc Hd Be Cf Dg Jj Ai Ia
To repeat the idea: every one of the 100 possible pairings of inner and outer square colours occurs exactly once. Each of the inner and outer colours occurs once in every row and every column.
You can swap any two rows or any two columns and it's still a Graeco-Latin square. I swapped rows around so that the blocks with the darkest outer colour lined up in a diagonal - if you call that one A, the result was
Ac Ga Bj Di Ff Ie Jd Hg Ch Eb Je Ad Ha Cj Ei Gg If Bh Db Fc Ig Jf Ae Ba Dj Fi Hh Cb Ec Gd Bb Ih Jg Af Ca Ej Gi Dc Fd He Hi Cc Ib Jh Ag Da Fj Ed Ge Bf Gj Bi Dd Ic Jb Ah Ea Fe Hf Cg Fa Hj Ci Ee Id Jc Ab Gf Bg Dh Df Eg Fh Gb Hc Bd Ce Aa Ij Ji Eh Fb Gc Hd Be Cf Dg Jj Ai Ia Cd De Ef Fg Gh Hb Bc Ii Ja Aj
Trying to meet any other design constraints by shuffling rows or columns around is next to impossible.
Pinning labels with these two-letter codes on them onto the blocks before putting them together made final assembly quite simple.
You can get Graeco-Latin squares for every size except 6x6.
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