The other day I received this email: “Fun fact: There are 65,534 tie-up combinations on a 16-shaft loom.” To which I replied, “And I bet there is someone out there trying them all.” Although I didn’t know the validity of the number at the time, I was pretty sure I was right that somewhere in the world a weaver was trying every tie-up combination, “just to see.”
In any case, that little exchange got me wondering how many treadle combinations there were on an 8-shaft loom, and whether 65,534 was the correct number of possible tie-up combinations on a 16-shaft loom. I could easily figure out that there were 14 possible combinations on a 4-shaft loom but had no idea how you could calculate the number of possible combinations for more shafts in any way that didn’t involve counting on your fingers and toes and adding columns of numbers. I tried a few ideas and got up to 64 combinations for an 8-shaft loom without even breaking a sweat, so I decided it had to be many more than that. After wondering for a bit about who I could ask, I contacted Tim McLarnan, of Tim’s Treadle Reducer fame, and asked him.
Tim's reply was wonderful and generous and made me wish I had had him for a math professor. He started at the beginning showing how you would calculate the number on 2 shafts, then 3 shafts and then how to apply the same principles to more shafts. Longer story short, each time you add a shaft, the number doubles. For calculating you need to leave in the two “useless cases” of all shafts or no shafts being tied up, but take them out afterward. For example the equation for 8 shafts is (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)–2, or (2 to the 8th)–2.
Fun fact: There are 2 possible tie-up combinations on a 2-shaft loom, 6 on a 3-shaft loom, 14 on a 4-shaft loom, 30 on a 5-shaft loom, 62 on a 6-shaft loom (I’ve never heard of a 7-shaft loom, but 126 on a 7-shaft loom), 254 on a 8-shaft loom, and yes, 65,534 on a 16-shaft loom. Want to know how many tie-up combinations there are on a 24-shaft loom? The equation is (2 to the 24th)–2. I’ll let you do the math, but the number of combinations is bigger than a breadbox, and someone out there is trying them all.