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COUNTING NINE PATCHES:
THE FIRST BILLION OR SO

Suppose you want to design a patchwork block on a 3x3 grid using only straight lines, no extra vertices, and at most three fabrics. How many possibilities are there within these restrictions? Whatever the total number may be, it's not hard to come up with at least a billion. The proof goes as follows:

Nine patch grid
Basic nine-patch grid

1. Consider first the nine component squares of the 3x3 grid. Each of these can be left whole and filled with any of the three fabrics (3 possibilities), divided by a right-slanting diagonal into two triangles filled with distinct fabrics (6 possibilities -- 3 fabric choices for the first triangle and 2 remaining choices for the second triangle), or divided by a left-slanting diagonal and similarly filled (another 6 possibilities). In other words, there are 3 + 6 + 6 = 15 ways to assign fabrics to each of the nine component squares. The total number of block designs produced in this way is therefore 15 to the 9th power -- something in excess of 38 billion.

Nine patch grid
Examples of the first 38 billion

2. Of course, these designs aren't totally distinct. If A and B are two designs in the series, it may be possible to obtain B from A by applying a sequence of flips and rotations to A or by permuting its fabrics. In fact, each block may appear in our grand series in as many as (but not more than) 48 guises. Flipping and rotating a block yields up to 8 variations (fewer if the block contains internal symmetries), and reassigning the fabrics produces up to 6 versions (fewer if not all fabrics are used) of each variation. To ensure that we don't count twice what may be the essentially the same design, we'll want to divide our 38 billion plus examples by 48.

same design twice
Designs that look different,
but aren't, really

3. No need to worry, however. There are plenty more designs left in the simple tic-tac-toe grid. We turn next to those designs obtained by picking two adjacent squares out of the grid, dividing this 1x2 rectangle with a diagonal, filling the resulting triangles with distinct fabrics, and then assigning fabrics to the remaining 7 squares of the grid as in paragraph #1 above. It's not too hard to see that there are 12 ways of picking two adjacent squares out of a tic-tac-toe grid, 12 ways of dividing and assigning distinct fabrics to the 1x2 rectangle, and (as before) 15 ways of assigning fabrics to each of the 7 remaining squares. Thus there are 12 x 12 x (15 to the 7th power) -- something in excess of 24 billion -- designs produced by this second raid on the great storehouse of nine patches.

Nine patch grid
Examples of the next 24 billion

4. If we add this 24 billion to the original 38 billion and divide the total by 48 we come out, obviously, with a billion and plenty to spare.

If only it were so easy to figure out which of the billion blocks to use where!

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