Rabbit Hole: Minimum L-seams

Sophia Wood examines Mrs. Perkins's quilt - the problem of dissecting an n×n square into the fewest integer-sided squares - through the practical lens of quilting: how many L-shaped seams (corners that prevent simple guillotine cuts) are unavoidable in optimal dissections. Framing L-seams as sewing challenges (and as cuts a paper cutter must handle), she defines a sequence a(n) giving the minimum number of L-seams for optimal quilts and computes values for n = 1..17, starting {0,0,0,0,0,0,1,0,1,0,1,1,2,1,1,2,2}. The exploration combines computational quilt catalogs with hands-on sewing considerations and raises concrete questions about larger n, the frequency of “lazy” (guillotineable) solutions, and whether longer corner seams (consecutive L-seams) occur.

The data show A(1)-A(6)=0; A(7)=1; A(8)=0 though among 28 dissections two require one L-seam; A(9)=1; A(10)=0 (75 dissections: 63 zero, 12 one); A(11)=1 (35 all with one); A(12)=1 (9 dissections: 8 with one, 1 with two); A(13)=2 (a 13×13 requiring two L-seams); A(14)=1; A(15)=1 (27 dissections, one with two); A(16)=2 (nine requiring two); A(17)=2 (two requiring two). The sequence has been submitted to OEIS and several open problems remain: which n need three or more L-seams, how counts scale with n, how many distinct minimal L-seam solutions exist, and whether related integer sequences are worth cataloging.