Math for the neckline – neck ribbing

Here's the math that I use to figure out how many stitches to cast on for the neck. 

If you don't like math, don't read this because a lot of it is in mathematical shorthand.

These calculations give a neck that is at least 120% the size of the person's neck (for comfort).  The lower case b, s, and f are for adjusting the number of stitches cast on to give some symmetry.  C = garment circumference times the gauge and N = person’s actual neck size times the gauge.  Then choose:

B = 30 % of N + 7% of C + b -- the number of stitches at the neck belonging to the garment back

S = 30 % of N - 7% of C + s -- the number of stitches at the neck belonging to the sleeve

F = B + f -- the number of stitches at the neck belonging to the garment front

For a simple crew neck, b = s = 0 or 1, and f = 8 or 10.

For a wider neck, change 30% to 35% or 40%. For a deeper neck, add a multiple of 4 to f.

However, to adjust for the ribbing (because I like symmetry):  For k1p1 ribbing, choose B, S, and usually F = 1 mod 2. (I.e., when divided by 2, the remainder is 1)

For k2p2 ribbing, choose B, S, and usually F = 2 mod 4.

For k3p2 ribbing, choose S = 2 mod 5 and B and usually F = 3 mod 5, or vice versa.

To take k2p2 ribbing to a k6p2 garment pattern, choose S = 2 mod 8 and B and usually F = 6 mod 8.  -- For the ribbed (k6p2) sweater in the last post, C = 160, N = 56, S =10 = 2 mod 8, and B = F = 30 = 6 mod 8.