I bet you thought you’d escaped algebra forever when you graduated high school, but these formulae just might help designers convert sock patterns to different sizes and gauges. Or knitters to design their own patterns. Most of this will be review for experienced sock knitters, but I thought it would be better to cover *all* the steps.

My examples will be shown with two sets of #’s: 64 sts and 50 sts TOTAL.

Notes/ key:

- H = # of stitches in the heel after all turning and decreases worked.
- F = # of heel flap sts; 1/2 of the TOTAL # of stitches.
- h = # of stitches
*in the heel*during turning. ‘h’ will be determined below.

**Heel flap:**

Ignore the above notations for the moment. *First* you have to knit the heel flap. Work 1/2 of the TOTAL stitches onto 1 needle (= ‘F’).

EXAMPLE:

64 sts TOTAL: F = 32 sts.

50 sts TOTAL: F = 25 sts.

If ‘F’ is an EVEN #, work the same # of rows for the heel flap. If ‘F’ is an ODD #, work (F – 1) rows:

- Row 1: Sl, work across (in chosen pattern).
- Row 2: Sl, purl across.
- Repeat Rows 1 & 2 until you have the designated # of rows.

**Turning the heel:**

Now comes the algebra. To determine how wide to make the ‘h’, look at ‘F’:

- if the ‘F’ is an EVEN #, choose an EVEN # for ‘h’.
- if the ‘F’ is an ODD #, choose an ODD #.
**Note: ‘h’ should be equal to approximately 10% of the TOTAL stitches (heel + instep), and when in doubt***round down*.

EXAMPLE:

F = 32. h = 6 (i.e. 64 X .10 = 6.4, which wasrounded downto 6). To place the heel correctly, add ‘F’ to ‘h’ and divide the whole by two: 1/2(F + h) => 1/2(32 + 6) = 19

- Row 1: K19, ssk, k1, turn.
- Row 2: Sl, p 7 (h + 1), p2tog, p1, turn.
- Continue working the heel in this manner: [knit to 1 stitch before the gap, work 2tog, work another st, and turn] until ALL heel stitches have been worked.
F = 25. h = 5 (i.e. 50 x .10 = 5). To place the heel correctly, add ‘F’ to ‘h’ and divide the whole by two: 1/2(F + h) => 1/2(25 + 5) = 15

- Row 1: K15, ssk, k1, turn.
- Row 2: Sl1, p 6 (h + 1), p2tog, p1, turn.
- Continue working the heel in this manner: [knit to 1 stitch before the gap, work 2tog, work another st, and turn] until ALL heel stitches have been worked.

There are 2 formulae for how many stitches will remain *after* turning the heel (this step is mostly important for writing patterns and converting sizes and gauge):

a:

H = h + 1/2[(F – h) – 2] + 2.

Use ONLY IF 1/2(F – h) is an ODD #: 1/2(32 – 6) = 13.EXAMPLE:

(Sock knit with 64 sts)

H = 6 + 1/2[(32 – 6) – 2] + 2 => 6 + 1/2[24] + 2 => 6 + 12 + 2 = 20.

H = 20 sts remain.b:

H = h + 1/2(F – h).

Use ONLY IF 1/2(F – h) is an EVEN #: 1/2(25 – 5) = 10EXAMPLE:

(Sock knit with 50 sts)

H = 5 + 1/2(25 – 5) => 5 + 1/2(20) = 15.

H = 15 sts remain.

**Gussets:**

To determine how many gusset stitches to pick up after turning the heel, divide the number of heel flap rows knit by 2 – which will correlate to the number of chained sts along each flap edge. If you want to avoid holes next to the heel and instep, pick up and knit 2 extra stitches, one on each end of the flap.

EXAMPLE:

32 rows/ 2 = 16: PU & K 16 sts (+ 2 if desired) along each heel-flap edge.

24 rows/ 2 = 12: PU & K 12 sts (+ 2 if desired) along each heel-flap edge.

Then work across the instep stitches and PU & K the same # on the other gusset. Decrease 2 stitches (one on either side of the instep) every other row until you have the original # of stitches CO.

There, how hard was that?

I just happened to see your link on Radar on Ravelry. It’s super! Thanks for putting it out there.

Illes can be used as a sex-neutral pronoun, like English ‘they’. ,

If you do not understand the answers, ask the doctor to explain them clearly. ,

Thanks so much for doing this post – I’ve found it really useful! And the heels worked out perfectly.

I was searching and searching for the math of a traditional heel for days! I’m making Christmas stockings and my pattern ended up needing twice as many total stitches as the pattern I was working to accomodate the intarsia letters I was putting on the top. THANK YOU SO MUCH!

Thank you for this! Being a mathematically illiterate knitter is no fun if you’re trying to design. I finally get it, thanks to your great explanation!

I just found this website…finally I can figure out turning heels on multiple socks with different stitch counts. You are a genius!