Updated November 2014

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The Diophantine Equation

The Magic Formula, also known as the Diophantine Equation, helps chart patterns in the area of necklines,
set-in sleeves, shoulder
shaping, increases and decreases and more. It is actually used very
often when charting a pattern as you rarely are able to have whole
numbers when dividing stitches into rows. A lot of the times you have a
left over, like 2.88 (the .88).

To help things along, I have put together a drop sleeve templates that you can download to use while you are reading this section.

**DROP SLEEVE TEMPLATE**

**SIDE NOTE:**

The actual name of the calculation is actually the "Diophantine Equation", it was called and now known also as the "Magic Formula" by ALLES.

To help things along, I have put together a drop sleeve templates that you can download to use while you are reading this section.

The actual name of the calculation is actually the "Diophantine Equation", it was called and now known also as the "Magic Formula" by ALLES.

As we all know, you cannot have a fraction of a stitch, because you
either have 1 (one) stitch or you do not. You cannot have 1/3 of a stitch or
.25 of a stitch. A stitch is a stitch. The Magic Formula gets rid of a
fraction, or a remainder less than one, and allows an even shaping across all rows. Speaking of fractions, I change any fractions into decimals, below is the fraction with it's decimal equivalent:

1/8th = .125

1/4th = .25

3/8ths = .5

5/8ths = .625

3/4ths = .75

7/8ths = .875

I am not so picky that I actually use all the decimals/fractions and keep it simple with 1/4 = .25 : 1/2 = .50 : 3/4 = .75. But if you are that thorough good for you!

The Magic Formula can be used for anything you want to knit and have increases or decreases evenly spaced on both sides of an item, either by necessity or preference. Even a dog sweater with stripes! I will try to explain the math that gets you there and actually do an example of the formula.

Or you can download a

In order to fully understanding the Magic Formula and pattern charting a must have is a booklet called

Also, Sandee's Kwik Knits has a CD that you can purchase that has 108 slides that will teach you how to chart garments using the Magic Formula. I personally have not used it, but I am sure it is a great resource.

Here is a link to Northtiptons site about the Magic Formula. All I can say is that she is a lot smarter at math than I am because I was L - O - S - T ! She converts inches to centimeters but for me it is easier to just knit a swatch and use centimeters as a measurement from the beginning.

But, just in case you need to convert inches to centimeters, you can do that with an online calculator that you will find on this link.

Now on to the "MAGIC"...

We will be using long division as a means to calculate how to knit up the sleeve and the sleeve cap.

1.This is the math that we use when we figure out how to increase and decrease over a certain number of rows evenly using the Diophantine Equation or Magic Formula.DIVISOR- The number you use to divide by

2.DIVIDEND- The number to be divided, the number you are dividing

DIVISORis 8

DIVIDENDis 24

8 can go into 24, 3 times, 3 is theQUOTIENT

3.QUOTIENT- The number of times theDIVISORfits into theDIVIDEND(it should be a whole number, if it is not use the Magic Formula)

DIVISORis 8

DIVIDENDIS 30

8 can go into 30, 3 times (3 x 8 = 24) with 6 left over, so 6 is theREMAINDER

4.REMAINDER- The number that is left over after theDIVISORis divided into theDIVIDEND.

90 stitches - 20 stitches = 70 stitches for increasing

BUT, you need to half this number as with a sleeve you increase on BOTH sides, otherwise you will get a lopsided sleeve. So, now you have 35 stitches.

70 stitches divided by 2 (sides) = 35 stitches

Also, in this example, you know that you have 140 rows to knit the sleeve increases, you know this by your swatch for yarn gauge, the arm measurement or the pattern. So, you take your rows of 140 and divide "BY" 35 stitches (35 each side is 70) and you get the number 4. Which is lucky because 140 is divisible by 35.

140 rows divided by 35 stitches = 4

This means you will add a stitch on each side on every 4th row until you reach 90 stitches, or 35 times. Simple enough!

But what happens when you divide rows and stitches but cannot get a whole number? What do you do with the leftover? That is where the Magic Formula comes in.

With the next example, #2, you need to look at it like a stack of building blocks. One calculation builds off of another one. I have also color coded some of the numbers so it is easier to follow along.

I will try my best to describe this equation using the Magic Formula...

Let's say you have a sleeve and you have 25 stitches at the cuff and 121 stitches at the shoulder sleeve area. You take 25 from 121 to find out how many stitches you need to increase, which is 96 stitches.

Then you divide 96 by 2 because you increase 1 stitch on each side (I always forget to half the stitches). So now you have 48 stitches.

Increase 1 stitch on each side 48 times

You know from your test swatch that

Uh Oh, now what?

Since you have a remainder of

So, you take

48 stitches can go INTO 136 rows

We then subtract the remaining stitches

Then add 1 (one) to the number of of how many times the 48 stitches can go into the rows,

You then state that you:

-Increase

- Increase

To double check your work...

2 rows @ 8 times = 16 rows

**2 stitches because you increase 1 each side, so you count 2

2 stitches @ 8 times = 16 stitches

Do not ask me how this works out every time, but it does. You just need to remember the steps, which numbers go where and in which order. In other words, there is a "direct relationship" with certain numbers and it always works the same way.

Off course, if you do not like doing math you can use any of the online calculators that I have listed on the Knitting Pattern Calculators page. But, if you do like fooling around with numbers, here is another explanation. I just found this fascinating to do.

As stated before, if your calculations come out even, there is nothing more to do. But when there is a remainder of rows or stitches, you need to use the Magic Formula for your knitting pattern. Now I will do a picture version....

**SIDE NOTE:** The top of the drop sleeve is the circumference of the person bicep plus the amount of ease that you added to the rest of the sweater. So, if you have 2" of ease, and the bicep was 17" = 19".

You have a drop shoulder, moderately tapered sleeves for your sweater. The dimensions are:

- Armhole 19"

- Cuff area 8"

- Length 20"

*SIDE NOTE: * ***when measuring a "Drop Down Sleeve" you measure from under the arm to the wrist, any other sleeve you measure from the shoulder to the wrist with a bent arm.

Your gauge from your knitted swatch is:

**6 stitches per inch**

**10 rows per inch**

19 inches x**6 stitches = 114 stitches at the top**

8 inches x **6 stitches = 48 stitches at the cuff**

and the sleeve is 20" tall x**10 rows = 200 rows total for the sleeve**

If you have decided to do a ribbed cuff of 2" this means that**:**

2" x 10 rows = 20 rows

Instead, I will subtract**5 rows** as I am knitting a crochet cast-on 4 times and then knitting 1 row, AND I will knit 3 rows of plain stitching and then casting off, which is **another 4 rows**.

**200 rows - (5 + 4) = 191 rows left to do the shaping**

**Shaping** - I then subtract the **48 stitches** that I cast on in stitches from the total at the top of the sleeve:

**114 - 48 = 66 stitches that I need to use for shaping**

I then take 66 stitches and ½ that number as my increases will be on each side of the pattern piece since it is a sleeve:

**66 stitches that I need to shape** **÷ 2 sides = 33 stitches**

So, I now have my numbers and I know that...

I cast on 48 stitches with a crochet cast on and knit 3 more rows of the crochet cast on, then 1 row of plain knitting. That is**5 rows** and **48 stitches**.

I now have to increase**33 stitches** over **191 rows** evenly across all rows.

I take 191 and divide by 33 and I get 5.78, so I know that I need to increase around every 5th row, but what about the .78???

You now switch to the Magic Formula/Diophantine Equation....

Click this link to download a cheat sheet on the Diophantine Equation

**Step 1:**

How many times can**33 stitches (DIVISOR)** go **INTO** **191 rows (DIVIDEND)**? Well you know about **5** **(QUOTIENT)** times since you already had figured that out. So **33 stitches** x **5** = **165 stitches with the remainder of 26 stitches (REMAINDER). **(This number must be less than the 33 stitches, or whatever number your calculations were, in this case it was 33).

**Step 2:**

Take the amount of times**33 (the stitches)** can go into **191 (rows)**, which is 5 and add 1 (one) to it. Whatever that number is, the **QUOTIENT, **you always add a 1 (one). That makes it 6.

**5 + 1 = 6**

Step 3:

You also need to subtract the**REMAINDER**, or remaining stitches (26) from the **DIVISOR**, which is **33 stitches** and you get **7 stitches**.

**33 stitches - 26 stitches = 7 stitches**

**Step 4:**

Your instructions for the sleeve increases:

- Increase each side of every**(quotient #)** **5th (the times 33 goes into 191) row**, **7** **times (divisor - ****remainder)**

- Increase each side of every**6th ****(quotient # + 1)** (the times 33 goes into 191, 5 + 1) row** ****(remainder)** **26 times** (the leftover stitches).

This happens because there is a direct relationship between numbers, and you should consider it as a step, a rule, to get your instructions correct.

**Step 5:**

Check your work:

**ROWS: I had 191 rows to use for the increases:**

5 rows x**7 times** = 35 rows

6 rows x**26 times** = 156 rows

**35 + 156 = 191 rows**

**STITCHES: I had to increase evenly using 66** stitches:**

**66 because you increase 1 stitch on each side

** **2 stitches **7 times** = 14 stitches

2 stitches**26 times** = 52 stitches

**14 + 52 = 66 stitches**

**
**

If you want to learn how to figure out a set in sleeve, there is a page for that in this section. It is much more complicated, so make sure you understand this page first. I hope that I have helped and not confused you even more! LOL

**Recommended Texts:**

*Charting by the Magic Formula published by Country Knitting of Maine*

or email her at: oth98@hotmail.com

*"KNITWEAR designs WORKSHOP, A comprehensive guide to hand knits"*

by SHIRLEY PADEN

Buy it at AMAZON.COM

As stated before, if your calculations come out even, there is nothing more to do. But when there is a remainder of rows or stitches, you need to use the Magic Formula for your knitting pattern. Now I will do a picture version....

You have a drop shoulder, moderately tapered sleeves for your sweater. The dimensions are:

- Armhole 19"

- Cuff area 8"

- Length 20"

Your gauge from your knitted swatch is:

19 inches x

and the sleeve is 20" tall x

If you have decided to do a ribbed cuff of 2" this means that

2" x 10 rows = 20 rows

but will reset my row counter back to zero once I have transferred all my rib stitches to the main bed. Always reset, it is easier.

Instead, I will subtract

I then take 66 stitches and ½ that number as my increases will be on each side of the pattern piece since it is a sleeve:

So, I now have my numbers and I know that...

I cast on 48 stitches with a crochet cast on and knit 3 more rows of the crochet cast on, then 1 row of plain knitting. That is

I now have to increase

I take 191 and divide by 33 and I get 5.78, so I know that I need to increase around every 5th row, but what about the .78???

You now switch to the Magic Formula/Diophantine Equation....

Click this link to download a cheat sheet on the Diophantine Equation

How many times can

Take the amount of times

Step 3:

You also need to subtract the

Your instructions for the sleeve increases:

- Increase each side of every

- Increase each side of every

This happens because there is a direct relationship between numbers, and you should consider it as a step, a rule, to get your instructions correct.

Check your work:

5 rows x

6 rows x

2 stitches

or email her at: oth98@hotmail.com

by SHIRLEY PADEN

Buy it at AMAZON.COM