Twill #1 answer

The finished swatch

The weave: DLLDLDDL / DLDDLDLL / LDLLDLDD / DDLDLLDL / LLDLDDLD / LDLDLDLD / DLDLDLDL / LDDLDLLD (L = light, D = dark, each row from left to right, top row first). Every row and column has 4 of each thread, and no thread runs three in a row.

A twill day swatch: Sunday's puzzle. The hardest idea it needed was ruling out a row that would copy another.

How to crack it: the first moves

1. A cross (x) joins two different threads, so this one (row 8, column 2) is dark.
2. Try every way to finish column 1 (half of each thread, never three in a row, its ties and crosses). Every one of them makes this square (row 1, column 1) dark.
3. Try every way to finish column 1 (half of each thread, never three in a row, its ties and crosses). Every one of them makes this square (row 5, column 1) light.
4. A tie (=) joins two squares of the same thread, so this one (row 5, column 2) is light too.
5. Two light threads sit side by side, so the square next to them (row 5, column 3) must be dark: never three in a row.
6. A tie (=) joins two squares of the same thread, so this one (row 6, column 1) is light too.

Every Twill swatch can be woven this way, one deduction at a time, without guessing.