| Math of hat decreases. |
[Nov. 2nd, 2005|05:34 pm] |
So, I'm making my little brother a Jayne hat for Christmas.
Since I like soft yarn, I'm making it in Knitpicks Merino Style, held doubled. My gauge is a bit small-"only" 4 stitches/inch. I think that using up *all* the orange and yellow I bought, minus enough for the pompom, will make a hat about the right size.
I know how many rounds I got out of the orange, and I wanted to know how many rounds I could get out of the yellow, given that I had to do decreases for it.
The decrease pattern is simple: decrease 7 sts every round until 7 sts remain, fasten off.
I realized that I could write down a surprisingly simple formula for how much yarn the decreases would take up, in terms of the amount of yarn a normal round takes up.
Pretend that the hat is made of 7 equal-sized pieces which are rectangular (during the "work even" rounds) and which are then triangular. Let n be the number of stitches in each of the even rounds. Note that n is equal to the total number of hat stitches divided by the number of decreases per round.
Then the triangular part has one n-1-stitch row, one n-2 stitch row, and so on.
Thus, the total number of stitches worked while decreasing is:
n-1+n-2+...+2+1=n(n-1)/2.
Thus, the total yarn required to do the decrease section is equal to the amount of yarn it takes to work (n-1)/2 even rounds.
If you only did decreases every other round, you'd need twice as much yarn.
Of course, this assumes that you need the same amount of yarn to work a k2tog as a knit stitch, which probably isn't quite true, but is probably pretty close. |
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