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Rabbit Hole: Minimum L-seams

Animation slicing a 13x13 quilt with two L-seams

I was trying to write a post about art with squarable numbers, and so I thought quilting a Mrs. Perkins Quilt might be a nice addition to the exploration. With 1500 words almost ready to hit send, I ended up down a rabbit hole. This post is rather long and has a lot of data, so bear with me, or skip it. I will have an artsy post with a quilt in the next week.

"For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches."

The solution looks like this:

A 13x13 quilt with two squares that sho

As I write this post, I have fabric at the ready to quilt this; but I have instead found myself going through many quilts to assess sewing technique. You see, I can't seem to find images of people sewing these quilts despite their name. It might be because they are a little tricky. I don't like sewing around corners - they can pucker (you might need to use a partial-seam trick). For this quilt, I will have to sew two L-shaped seams (shown in yellow dashes above).

{ 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, ... }

  • What quilt will require 3? 4?
  • What is the distribution of lazy quilts vs. harder quilts for each n as it gets larger?
  • Will two L-seams ever be needed in a row (like a U-seam)?
  • What other sequences can spin off of this one?
  • Is it worthy of being an OEIS sequence, or is it too niche? (I called a fellow puzzler and colleague for this one in my uncertainty)

a(2) = 0

A 2x2 quilt cut into four size one squares.

a(3) = 0

A 3x3 quilt split into 6 squares where a 2 size is surrounded by 1 sized squares.

a(4) = 0

A 4x4 quilt split into 7 blocks symmetrically with 2 sized two on the top then a 2 centered in the middle surrounded by 1's stacked on the sides.

a(5) = 0

A 5x5 quilt split into 8 squares of sizes 1,2, and 3 that can be sliced up by guillotine.

a(6) = 0

A 6x6 quilt split into 9 squares of sizes 1,2, and 3 that can be sliced up by guillotine.

a(7) = 1

Four 7x7 quilts with 9 blocks all requiring an L-seam.

a(8) = 0

28 quilts ( 26 need 0, 2 need 1) (only one zero is shown)

Three 8x8 quilts where the one on the right can be sliced without L-seams and the two on the left require one. the blocks are sized 1-4, but only the L-seams have size 5 blocks.

a(9) = 1

Nine quilts of size 9x9 that need one L-seam.

a(10) = 0

75 quilts ( 63 need 0, 12 need 1) (only one zero is shown)

a(11) = 1

35 quilt of size 11x11 that all have one L-seam

a(12) = 1

9 quilts (8 need 1, 1 needs 2)

Nine quilts of 12x12 with 8 that have one L-seam and one that has two. The L-seams for the 1 are around rectangles of multiple blocks.

a(13) = 2

A 13x13 square split into 11 squares with two L-seams needed in the top left and lower right blocks.

a(14) = 1

a(15) = 1

twenty-seven quilts, where one has tow L-seams and the rest have one.

a(16) = 2

nine 16x16 quilts all needing two L-seams

a(17) = 2

two 17x17 quilts that look very similar needing two L-seams

References