I posted the following on Facebook: Here's a project for someone ... A patchwork quilt based on the 26 solutions of Langford's Problem for N=7 (26 rows, 14 columns). Only the first 11 rows are shown here.
You can see all 26 rows [HERE]
The first Langford Quilt - 26 Solutions for Pairs 1..7
A neighbor, who is a quilter, was inspired by the colored swatches in the Langford Machine animation, and so we had started talking about quilt patterns. I'd done such a coloring by hand a long time ago, but I have not found it in my files yet. So, I generated the above pattern and posted it on Facebook.
The quilter said that the above pattern was boring, static.. not dynamic. She said it doesn't show movement like the Langford Machine... Also, she said, a quilt would have black between the colored patches to make them stand out. So I experimented, and quickly realized that if there were no black squares between the rows, the colors would 'run together' when a color was in the same position from one row to the next.So, here we have my first satisfactory Langford Quilt:
See below for an observation about colors running together.
20 Solutions for triplets of 1..10
This quilt represents the 20 solutions using triplets and Nickerson's Variation of Langford's Problem. Nickerson's rule is that a given triplet for 'k' has k-1 other squares appearing between the successive occurrences of the three 'k' squares. Consider the following numerical solution, where I show the three 10's with 18 other numbers 'among' them.
There are 20 such unique solutions. In the Quilt, Each row has 30 squares, 3 of each of 10 colors, and they are spaced accordingly:
Lt Gray represents the 10's, with 9 squares between each Lt Gray square. The same is true for all smaller numbers 9, 8, 7, 6, 5, 4, ... the 3 yellow squares have two squares between them; the 3 orange squares have one other square between them; the 3 red squares have NO squares between them. (Find the three 1's together in the above numerical solution.)
Here is the quilt, for your appreciation.
33 Solutions for triplets of 1..11
This Langford Quilt is based on 33 solutions of a variant (using triplets instead of pairs) for n=11. You can see the triplets if you look for them. The 33 solutions are shown, one per row.
The variant is also "Nickerson" -- i.e. the three 1's appear TOGETHER, not separated by one block each. The 2's are separated only by 1 block, not two. And so on.
Note that while this is 33 rows of 33 colored blocks, there are 32 BLACK vertical bars that make the pattern twice as wide as it is tall.
Which Sets of solutions make good quilts?
You can see in the table why I chose 7 pairs for one quilt (26 solutions of 14 blocks) and 11 triplets of a variant (33 solutions and 33 blocks!) and finally, the 20 solutions for 10 triplets.
You can examine the whole table on the Langford site [HERE].
Visual Interest created by Colors Running Together
When a pair or triple of blocks stay in the same position from one solution to the next, their colors will run together for those rows. However, colors can also run together if just one of the blocks lines up with a block of the same color in an adjacent row, as shown by the red and orange bars in the following portion of the L(2,7) quilt.
Try finding these runs in the large quit images above (pairs and triples). This is an interesting aspect of Langford's Problem that might be fun to explore.
