Few threads
Five 7×7 boards and three 8×8 ones, each with only 4 or 5 pairs of buttons.
On a 7×7 board that means some threads run for a dozen cells or more. The buttons often sit close together, yet the thread between them has to take the long way round to cover the cloth. Short, direct connections are almost always a trap.
Start from the buttons near the border and corners: their threads have the fewest choices. Ask which thread could possibly reach a far empty corner, and you often find there is only one. More in long threads and corners and edges.