Inside the logo: a puzzle hiding in the quilt

The Quiltoku logo is a ten-by-ten grid of squares within squares, copied patch by patch from a real quilt. Look closely and you find that the quilt is also a perfect Double puzzle, with a smaller, complete puzzle sitting in one corner.

Ten colours, two layers

Every cell has an outer square and an inner square. The outer squares use the ten colours of the quilt, and so do the inner squares. Now look at how they are arranged:

• Each colour appears exactly once in every row and exactly once in every column of the outer squares.
• The same is true of the inner squares.
• No pairing of an outer colour with an inner colour is repeated anywhere. There are 100 cells and 100 different pairs, so every possible pair appears exactly once.

The first two points make each layer a Latin board, the kind described on the Latin page. The third point is the special one. It means the two layers are orthogonal: together they form a Graeco-Latin square of order ten.

Why ten is the interesting number

Ten is one of the sizes that the mathematician Euler believed could never have such a square. He studied the case of six and could find nothing; he guessed the same would hold for ten, fourteen and every size like them. It was a reasonable bet. It was also wrong, and the proof arrived in 1959 and 1960, when the sizes he doubted were shown to have solutions. The Latin page tells that story. The point here is simple: building a ten-by-ten pair of this kind is hard, and there it is, in fabric.

We do not know whether the quilt was planned as mathematics, and we would not claim it was. But the arrangement is there, and it is exact. When the logo was copied into the game, two patches turned out to be wrong, and they were checked against the real quilt. Once those two were put right, the grid became a true Graeco-Latin square, which is a good sign that the copy matches the original.

A complete puzzle in one corner

Here is the part that is easy to miss. Take the bottom-right three-by-three block of the logo.

It uses only three of the ten colours. In the outer layer, each of the three colours appears once in every row and once in every column. The same is true of the inner layer. And the nine cells hold nine different outer and inner pairs. That makes this tiny block a finished Double puzzle on a 3×3 board: a Graeco-Latin square of order three, sitting in the corner of the order-ten one.

Of all the blocks you could cut out of the logo, from 2×2 up to 9×9, this is the only one that is complete in both layers. The rest of the quilt does not break into smaller puzzles. That makes this corner a small coincidence, or a small signature, depending on how you like to read it.

What would it look like as a puzzle?

Remove a few squares from the 3×3 and you have a very easy Double puzzle: the missing marks can each be found from the row and the column alone, in both layers. Remove squares from the whole 10×10 and you have a hard one: the quilt as a puzzle, with some of the patches left for you to find.

Where the colours come from

The ten colours are burgundy, lemon, red, black, navy, mint, cerulean, sage, pale lavender and periwinkle, matched to a photograph of the fabric. Burgundy and navy are the darkest of them. Because some pairs, such as navy and black, or burgundy and red, are easy to mistake for each other, the stitched Quilt marks give them their own thread colours. You can read about the quilt itself, and the person who made it, on the quilt page.