Latin squares

I often spend a long time trying to work out the ‘optimal’ placement of my quilt blocks when I’m at the assembling stage. While I was trying to assemble the blocks for my brother’s birthday quilt, I began to wonder if I could apply some maths. Usually, I have an n x n arrangement with n blocks of each of n colours (where n is any integer). I then aim to have each colour only once in each row and each colour, for example, as in the first quilt I ever made. I’ve overlaid numbers on each colour to make the arrangement clearer.

It turns out that this requirement is that of a Latin square, as studied by the Swiss mathematician Leonhard Euler.

Another example is my Eschereque cushion, which I didn’t end up quilting, but it’s still arranged using a Latin Square arrangement of four colours.

These two examples are quite simple in that there is only one variable, the colour of the block. Things could be made more complicated by having four colours and four designs of block, for example. In this case, I’d want to have each colour once and only once in each row and column, and each design once and only once in each row and column. I haven’t yet made a quilt that follows these requirements, but I think that’ll be another one for the list! Squares where there are two (or more) classes of variable, where the elements in each class are arranged in Latin squares (i.e. every row and every column contains each element only once) and each cell is unique are known as Graeco-Latin squares.

Euler spent some time on this problem, and worked out that it was impossible to create a 6×6 Graeco-Latin square. You might remember I was planning to make a 6×6 block quilt top for my brother’s quilt, and I wanted to arrange the blocks so that the outer colours were in a Latin square and the inner colours were in an orthogonal Latin square of their own – but as this is impossible, I’ve revised my plan and have gone for a 5×6 arrangement. This means that in each column, there is one instance of each of the six outer colours. I also relaxed the requirement on the inner colours to be in a Latin square of their own, so my brother’s quilt isn’t a Graeco-Latin square, let alone a Latin square – it’s not even a square, after all!

However, I’m pleased to have discovered the existence of Graeco-Latin squares. I won’t try to design any 6×6 Graeco-Latin quilt tops in the future either, but I’m really pleased to discover that a 5×5 Graeco-Latin square for four classes exists.

How else do other people arrange their quilt blocks? Would anyone be interested in a pdf download of Latin and Graeco-Latin squares for quilt arrangements?

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