Rabbit Hole: Minimum L-seams
Recorded: Sept. 18, 2026, 3:08 p.m.
| Original | Summarized |
Rabbit Hole: Minimum L-seams M Home Sign in rabbit hole Rabbit Hole: Minimum L-seams Sophia Wood 16 Sep 2026 Share 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. Please note, this post will get updated as I work through the problem. The update on the quilt is here. The quilt:Ok, first off, Mrs Perkins' quilt is a lovely math problem. The challenge is to slice a square of side n into smaller squares with integer side lengths, use the fewest squares possible.The website squaring.net has a lot more on the problem, including the original wording, and Wolfram has a demonstration to explore. "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."Henry E. DudeneyThe solution looks like this: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). If sewing together doesn't make sense, another way to think about this is if you have a paper cutter, how can you minimize the number of corners you have to cut so that you can guillotine cut the rest. This got me thinking: for any optimal Mrs. Perkins' Quilt of side n, what is the minimum number of L-seams required? And this is where we start building a sequence a(n):{0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, ... }Below is the exploration for n values from 1 to 17. I started looking at the larger numbers, but there can be a lot of quilts, and I honestly want to get sewing on this 13x13 and maybe a couple of others. Below is just data, then references. Some questions were also pondered, but not answered: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?How many L-seam solutions exist for each quilt? 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) Data:a(1) = 0 - it's just a square.a(2) = 0 a(3) = 0 a(4) = 0a(5) = 0 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) = 028 quilts ( 26 need 0, 2 need 1) (only one zero is shown)a(9) = 1 a(10) = 0 75 quilts ( 63 need 0, 12 need 1) (only one zero is shown)a(11) = 1 a(12) = 1 9 quilts (8 need 1, 1 needs 2)a(13) = 2 a(14) = 1 a(15) = 1 a(16) = 2 a(17) = 2 ReferencesData used: Stuart E. Anderson, "Mrs. Perkins's Quilt," Squaring.net — quilt data computed and compiled by Ed Pegg Jr, Stuart Anderson, James Williams, Lorenz Milla, Ed Wynn, Richard Guy, and others.Henry Ernest Dudeney, "Mrs. Perkins's Quilt," Problem 173 in Amusements in Mathematics, Nelson, London, 1917 (reprinted by Dover; free text at Project Gutenberg).Eric W. Weisstein, "Mrs. Perkins's Quilt," MathWorld — A Wolfram Resource."Squaring the square," Wikipedia.Ed Pegg Jr, "Math Games: Square Packing," MAA Online, December 1, 2003.Ed Wynn, "Exhaustive generation of 'Mrs Perkins's quilt' square dissections for low orders," arXiv:1308.5420 [math.CO], 2013–2014.OEIS A005670: Mrs. Perkins's quilt — smallest coprime dissection of an n × n square.TBD - maybe a new OEIS sequence?? Submitted for review. Sign up for more like this. Enter your email Mrs Perkins' Quilt After a rabbit hole of playing with the minimum L-shaped seams of Mrs Perkins' Quilts, I submitted the related sequence to the Online Encyclopedia of Integer Sequences (OEIS) for consideration. Then, I washed, pressed, cut, sewed, ripped stitches, then sewed, then pressed a mini-quilt for the 13x13 18 Sep 2026 Mathober 2026 Prompts Are you ready? October is right around the corner and you know what that means? Mathober! This years prompts are on the website mathober.com and below. If you are new to Mathober, no worries, you can jump right in. Each day there is a word or phrase for you 10 Sep 2026 Sci Art September It's time for SciArt September hosted by Kristin Henry (vis.Social/@kristinHenry) with #SciArt on Mastodon. This year, I made wallpapers. There are 137 images per doodle that are formed from mirroring, gliding, and rotating an image. I am hosting on SmugMug because of the sheer volume of 03 Sep 2026 Fractal Kitty © 2026 Powered by Ghost I <3 playing with math at the Recurse Center! Sophia |
The exploration detailed in the text centers on finding the minimum number of L-shaped seams required to construct an optimal dissection of a square of side n into smaller squares with integer side lengths, framed by the problem of Mrs. Perkins' Quilt. The underlying mathematical challenge involves determining the minimum number of square portions a quilt of side n can be composed of, which is related to the original problem posed by Henry E. Dudeney. The text posits that this minimization translates into finding the minimum number of L-seams necessary when piecing together the constituent square pieces. The author investigates this by defining a sequence, denoted as a(n), which tracks the minimum number of L-seams required for various values of n. The initial exploration, covering n values from 1 to 17, generated a sequence: {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2}. This sequence arises from analyzing how an n by n square can be sliced into smaller squares, particularly considering constraints related to minimizing corners that require cutting. Specific data points illustrate this complexity. For instance, for a 6x6 quilt, the minimum number of L-seams is zero, while for a 7x7 quilt, the minimum is one. As n increases, the dependency on the number of seams changes, as seen in the $12 \times 12$ case where the minimum is one, and for the $13 \times 13$ quilt, the minimum number of L-seams is two. The author notes that this investigation prompts further mathematical questions regarding the distribution of "lazy" versus "harder" quilts as the dimension n grows, and whether patterns, such as consecutive U-seams, can occur. The text references existing mathematical work related to square dissections, including work by Stuart E. Anderson and the compilation of data by researchers such as Ed Pegg Jr. and others. The sequence generated by this study has been considered for inclusion in the Online Encyclopedia of Integer Sequences (OEIS) under the identifier A005670, which relates to the smallest coprime dissection of an n by n square. The exploration emphasizes the relationship between geometric tiling problems and sequence analysis, seeking a deeper understanding of the combinatorial minimization inherent in partitioning a square into squares. |