Taking the magic out of the "Magic Formula"

This is the first in a series of posts explaining the use of the "magic formula", a simple mathematical equation which allows knitters to calculate evenly spaced shapings in varied situations. I'm putting this here as much for my own reference as anything else. I'll be expanding upon the varied uses of the MF as soon as I get the chance to put fingers to keyboard. I'm a busy boy, so it'll probably take me a while to write the complete series. Please be patient....

What exactly is the magic formula?
The “magic formula” is a nickname coined by Alles Hutchinson, an American machine knitting expert, for a mysterious sum she discovered being used frequently in 1970's Japanese knitting magazines. This strange, but useful equation allowed knitters to precisely calculate their garment shapings, eliminating guesswork or trial and error. Although not immediately clear as to why, it worked perfectly every time as if by magic, hence the name.

The “magic formula” is more correctly known as a “diophantine equation” and was invented by Diophantus, an ancient greek mathematician credited as the father of algebra. A diophantine equation is defined as an equation in which only whole number solutions are allowed. In other words, the answer to the sum can only be a whole number, no fractions or remainders are allowed. This is exactly what makes it so useful to us as knitters, bearing in mind that we can't work with fractions of stitches or rows, only whole ones.

There are 4 basic uses or expressions of the magic formula:

Expression 1a - used to evenly space shapings along a diagonal line, such as on a sleeve underarm, a raglan armhole or a v-neckline.

Expression 1b - same as above, but shapings always occur on even rows only. This is particularly useful for handknitters who prefer to work shapings with the RS of work facing. For this particular expression of the formula to work, there has to be at least two times the number of rows as stitches.

Expression 1

Expression 2 - used to evenly divide up groups of stitches for shaping, such as on a sloped shoulder or flared panel shapings on a sideways knit skirt. Shaping can only happen on alternate rows with this particular expression of the magic formula.

Expression 2

Expression 3 - used to evenly space shapings horizontally across a single row of knitting, such as in a circular yoked sweater, or to deliberately create fullness above a ribbed border.

Expression 3

Expression 4 - used to evenly space buttonholes or other design elements such as cable panels across a knit piece.

Expression 4

I'm working on diagrams to explain everything as clearly as possible. So stay tuned!