Craft maths 1: Combinations
I started to crochet an afghan the other week and decided that I wanted it to be about 15 by 15 hexagonal motifs, using three colours per motif and preferably using every combination of the colours in the set. I could tell that 15 would be a good width for the blanket, thanks to a test swatch and a tape measure.
Thanks to remembering some O-level maths, I knew the best-fit solution would be easy to work out.
To determine the number of colours you need to get a certain number of shapes using a prescribed number of colours, you will be using a Combination. This Combination will tell you the number of different mixes of colours you can make by drawing a certain number out of a larger set.
The number of colours = n, the prescribed number of colours = r, and the formula for the Combination is
nCr= n!
r!(n-r)!
n! means factorial: for n, the 'product of all positive integers less than or equal to n', according to Wikipedia.
It's much easier than it sounds: 1! = 1
2!= 2x1= 2
3!=3x2x1=6
4!=4x3x2x1=24
5!=5x4x3x2x1=120
6!=6x5x4x3x2x1=720
7!=7x6x5x4x3x2x1=5040
8!=8x7x6x5x4x3x2x1=40,320
And if you don't like multiplying, you can just type 8! into Google and it will supply the answer for you.
We know that r=3, because each hexagon has three colours. My ideal was for a blanket 15x15, or 225 motifs. I know that each combination of three colours can be made in six ways (3!), because there is a centre, middle and outside for each hexagon. So the combination of red, white and green can be: RGW, GRW, RWG, WRG, GWR, WGR. Dividing 225 by 6 gives me 37.5, so nCr should equal as close as possible to 37.5.
Obviously everything up to 4! is too small a number, and 5! is, too, because even just dividing it by r! (6) gives us 20, which is too few. Doing the whole calculation gives us 10, way too few!
If we run the numbers for 6! we get 720 divided by (6(6-3)!), which is 720 divided by 36, or 20. Not enough.
For 7!, 5040/(6x4!) = 35
This might work, 35x6=210, which is 15x14, just one row smaller!
For 8!, we have 40,320/(6x5!) = 56
This is probably too many options, 56x6=386 motifs, which is a bloody enormous rug, 15 motifs by 27.3 rows! (NB exclamation, not factorial.)
So seven colours and 210 hexagons it is!
Maths: the real cornerstone to effective craft. Fingers crossed I will have time to write up the circle geometry notes before the year is out.
Thanks to remembering some O-level maths, I knew the best-fit solution would be easy to work out.
To determine the number of colours you need to get a certain number of shapes using a prescribed number of colours, you will be using a Combination. This Combination will tell you the number of different mixes of colours you can make by drawing a certain number out of a larger set.
The number of colours = n, the prescribed number of colours = r, and the formula for the Combination is
nCr= n!
r!(n-r)!
n! means factorial: for n, the 'product of all positive integers less than or equal to n', according to Wikipedia.
It's much easier than it sounds: 1! = 1
2!= 2x1= 2
3!=3x2x1=6
4!=4x3x2x1=24
5!=5x4x3x2x1=120
6!=6x5x4x3x2x1=720
7!=7x6x5x4x3x2x1=5040
8!=8x7x6x5x4x3x2x1=40,320
And if you don't like multiplying, you can just type 8! into Google and it will supply the answer for you.
We know that r=3, because each hexagon has three colours. My ideal was for a blanket 15x15, or 225 motifs. I know that each combination of three colours can be made in six ways (3!), because there is a centre, middle and outside for each hexagon. So the combination of red, white and green can be: RGW, GRW, RWG, WRG, GWR, WGR. Dividing 225 by 6 gives me 37.5, so nCr should equal as close as possible to 37.5.
Obviously everything up to 4! is too small a number, and 5! is, too, because even just dividing it by r! (6) gives us 20, which is too few. Doing the whole calculation gives us 10, way too few!
If we run the numbers for 6! we get 720 divided by (6(6-3)!), which is 720 divided by 36, or 20. Not enough.
For 7!, 5040/(6x4!) = 35
This might work, 35x6=210, which is 15x14, just one row smaller!
For 8!, we have 40,320/(6x5!) = 56
This is probably too many options, 56x6=386 motifs, which is a bloody enormous rug, 15 motifs by 27.3 rows! (NB exclamation, not factorial.)
So seven colours and 210 hexagons it is!
Maths: the real cornerstone to effective craft. Fingers crossed I will have time to write up the circle geometry notes before the year is out.