Maths-brained Mumsnetters help!

[Broomfondle]

Our house is currently struggling with a maths query.
I've just finished knitting a baby blanket and out of interest asked my DH if there was a formula for working out how many stitches I'd done in total.

I knitted the blanket on the round which means you go round and round in a big spiral on a circular set of needles. The number of stitches increased by 2 on every corner, every other row. This actually makes the shape a square.

The very 1st centre row had 4 stitches (total). Every other row is a repeat of the one before with no increase. Then the next row increases by 2 stitches each corner. So first few rows are:

1. 4 stitches total (1 stitch per side of the square)
2. 4 stitches total
3. 12 stitches total (3 stitches per side of square)
4. 12 stitches total
5. 20 stitches total (5 stitches per side of square)
6. 20 stitches total

Etc

So simply by adding 4+4+12+12+20+20 I know there are 72 stitches total in the first 6 rows.

What I asked my DH was is there a formula for working this out?

The final row is a non-increasing row with 127 stitches per side, 508 total.

We have been up with the baby in the night now poor DH is in his dressing gown scratching his head over a piece of paper.

I said you could treat it as increasing by four stitches every row rather than 8 every other row. DH said this would give you the answer but not the formula, as it would only be correct half the time (depending on whether you finished on an increasing or non-increasing row).

I just wonderer how many stitches I'd done total but have now set DH off and he's scribbling down x this and y that etc.

Can anyone help?

This is called an arithmetic series. Supposedly, Sir Isaac Newton figured this out as a schoolboy when his teacher (hoping for a quiet morning) told the boys to add all the numbers from one to five hundred. Newton noticed that if you start from both ends and work inwards, 1+500=501, 2+499=501, 3+498=501, etc., and there are 250 such pairs - so the answer is 501 x 250.

Looking only at the odd numbered rings, you want to calculate 4 + 12 + 20 + ... + 492 + 500 + 508. You know that each number is eight larger than the previous one, so not counting the four there are (508-4)/8=63 numbers, which is 64 numbers including the four. Using Newton's trick of working inwards from the ends, you can see that there are 32 pairs of numbers (4+508, 12+500, 20+492, etc) that each add to 512. So your answer is 32 x 512 = 32,768 stitches in the odd numbered rings. Obviously there are the same number of stitches in the even numbered rings, for a grand total of 65,536 stitches overall.

I'm assuming you have two rings of 508 stitches at the end there, although I may not have understood quite right. If you only had one ring of 508 stitches then it's 65,028 stitches total.

I think it is...

If n is your total number of rows the total stitches is:

4 x n x (n+1)/2 - 4 x n/2

My thinking (in case I’m wrong but this helps) is the rows nearly follow totals of row number x 4. The odd rows do but the evens don’t.

So you need to start with a formula like:

(1 x 4) + (2 x 4) + (3 x 4) + ... + (n x 4)

This can be rearranged as 4 x (1 + 2 + 3 + ...+ n)

Which can be written as 4 x n x (n +1)/2

You then need a deduction item for every other row of four stitches (eg row two out formula is currently doing 2 x 4 = 8 but you have 4 in reality). This gives rise to the last item: 4 x n/2 (ie take off 4 for half the number of rows).

You can simplify 4 x n x (n+1)/2 - 4 x n/2 to

2n x (n+1) - 2n

Which simplifies again to 2n x (n + 1 - 1)

Which is 2n x n

So surprisingly simply it’s 2 x (n squared)

Let me know if you think that works. It does for your 6 rows.

I thought it was Gauss?

The formula for adding up the a series like this is

Total = n/2 times (2a+(n-1)d)

Where n = number of numbers
a = first number
d = common difference.

@dootball - according to Wikipedia you're right, it was Gauss. I'm sure I was told Newton, but it was the better part of thirty years ago. The maths works either way!

...although Wikipedia also says that Gauss only re-invented the method, and it is found in the work of an Indian mathematician called Aryabhata, from 499AD.

Supposedly, Sir Isaac Newton figured this out as a schoolboy when his teacher (hoping for a quiet morning) told the boys to add all the numbers from one to five hundred. Newton noticed that if you start from both ends and work inwards, 1+500=501, 2+499=501, 3+498=501, etc., and there are 250 such pairs - so the answer is 501 x 250

That would be an interesting experiment to do in a class of pupils nowadays - assuming they hadn't ever come across the formula.

So long ago.
I know to add numbers going up by 1 it is quiet easy.

N/2 (N+1)
Say you have the numbers 1 to 100.
The N + 1 means your just adding 100 and 1 together to get 101.
You know there are 50 pairs so you times
101 by 50....

So we need to know how many rows you have to knit.

I like the handshake problem.
If there are 10 people and they all shake hands, how many handshakes are there?
What if there were 50 people?
What if there were X people...

It might need to be updated for Covid - 19 though.

Newton noticed that if you start from both ends and work inwards, 1+500=501, 2+499=501, 3+498=501, etc., and there are 250 such pairs - so the answer is 501 x 250

That would be an interesting experiment to do in a class of pupils nowadays - assuming they hadn't ever come across the formula.

You don't need to be Newton to come up with this. In my early twenties someone set this as a puzzle and I came up with the formula. My friends were convinced I must have come across it before, but I hadn't, and I'm really no mathematical genius though more so than my friends I guess. OK I was older than Newton but I imagine at the age of 8 or whatever he could easily have trounced me in a maths contest.

I think TimeToParty has it for even numbered rows, but not for odd numbered rows - and possibly not for the final row if it is even and none increasing.

I think total stitches in the square for an odd number of rows is 2n^2 +2
Total stitches in the square for an even number of rows is 2n^2

Someone check my maths tho!

@RedCatBlueCat - your formulae check out. If you arrange the OP's numbers into two squares, the formula can be seen fairly quickly (at least for even n).

@Broomfondle - can you show us a picture of the pattern, as I am not quite sure I can visualise how your stitches are laid out.

Oh, and the story is about GAUSS not NEWTON, adding up first 100 positive integers when aged 7.

Thank you so much everyone for taking time out on a Sunday to try and help us with this!

@SpeverendRooner Thank you for the answer to the total number and the little bit of history! DH knew this way but (if I'm understanding correctly) it's a way of getting the answer but not strictly a formula due to the non-increasing (even) rows?

@TimeToParty and @RedCatBlueCat I think you've both got it. Where DH was head scratching was he was trying to come up with one formula, he said it didn't sit right with him that you had to use two (). I think he was saying you'd have to know the number of rows and he felt there should be a neater way of doing it with the information we had but I think logically it can only be done with two?

The only trouble now is I've taken the total number of stitches (65,536) and plugged it into the formulae to try and work out how many rows I've done.
I think I've done 128 (using @SpeverendRooner 'pair' of numbers as equivalent to a pair of rows) but when used in the formula I get half the total number of stitches (32,768) despite doubling for odds/evens.

Who knew this would be so complex!

I've attached a pic, it's not really the knitting pattern but it's a way of showing the distribution of stitches (the little dashes).

Thank you again everyone, I love the wisdom of Mumsnet. Any more comments on this welcome!

Especially any working showing how to get the number of rows working backwards from total number of stitches.

For a whole number of even rows, the number of rows will equal the square root of (stitches / 2). But for 65,536 stitches, sqrt(65536 / 2) doesn't give a whole number. It does for 32,768 (--> 128 rows).

Thanks @DadDadDad
So is the total number of stitches actually 32,768?

Did you come up with 65536 by doing 128 multiplied by 2 then squaring? 2n^2 means you need to do the squaring first (powers come before multiplying), so square 128 > 16384 then multiply by 2 > 32768 stitches.

I dont get a "nice" answer of rows for that number of stitches, but it is just over 180 rows fir that number of stitches.

For a given number of stitches, S
If even, number of rows=square root(S/2)
If odd, number of rows= square root ( (S-2)/2)
And one should give a whole number!

So is the total number of stitches actually 32,768?

If you did 128 rows then yes (with 127 stitches on each side of the final row).

So you need to know the total number of stitches at the end rows and the number of rows.
So long as you know one answer you can find the other .
The 4 is there as it's your cast on number.
Because you do an increase none increase you have twice the rows 128.

The second bit is the Gauss's formula.
Stitches 1st row + stitched last row.
This number is times by number of rows then you half the answer, as you have that many pairs.

@DadDadDad I got 65,536 from first reply to my post:

Looking only at the odd numbered rings, you want to calculate 4 + 12 + 20 + ... + 492 + 500 + 508. You know that each number is eight larger than the previous one, so not counting the four there are (508-4)/8=63 numbers, which is 64 numbers including the four. Using Newton's trick of working inwards from the ends, you can see that there are 32 pairs of numbers (4+508, 12+500, 20+492, etc) that each add to 512. So your answer is 32 x 512 = 32,768 stitches in the odd numbered rings. Obviously there are the same number of stitches in the even numbered rings, for a grand total of 65,536 stitches overall

Like PP I get a non-whole number of rows (181 and a bit) using this number but get what I think is the correct number of rows using 32,768. However I'm not sure what's the right total number now - 65,536 makes sense using Gauss's way of doing things I think, but doesn't come up right if worked backwards from using redcatbluecats formulae

Thanks @mummmy2017 I'll sit down with a bit of paper and follow what you did

The error in that is that actually 32 x 512 = 16,384, so that's why the answer is half what you were getting.

Aha! So knowing that I ended on an even row and using the formulae:
2n^2 +2 (odd)
and
2n^2 (even)
I can work out that I have done 32,768 stitches total over 128 rows.
Thank you all! So much! (And a nod to Gauss)

I so cheated.
I got my A level child to do it..
So the first bit works. On all rows...
8n-4= stitches...
8 by 1 row -4 = 8 - 4 = 4 stitches.
8 by 2 rows -4 = 16 - 4 =12 stitches...
8 by ? Rows -4 = 8 * ? - 4 = 508 stitches
We add 4 on each side to get rid of the - 4.
8 * ? = 512 so 512 /8 = 64...
But because we know there is a second row with the same number of stitches we have to double our answer....
The row with no increases always comes after the increase rows. So it will always be an even row.
So. We know we have 128 rows or 64 pairs.
We do the Gauss's equation.
Add the first number and the last numbers.
4 + 508 = 512.
We then just do 64 lots of 512.....
32768.