Since you can start with 1 and add it to itself to get 2, add again to get 3, and again to get 4... we say that 1 is a generator of the set

In fact it is an example of something that mathematicians call a group

It is a group of numbers and, as we've seen, you add them modulo 24, so eg. 15 + 20 = 35 = 13

Taking n = 24 and a = 9, the nails are {0, 9, 18, 3, 12, 21, 6, 15}

Remember, these are the numbers {0, 9, 18, 27, 36, 45, 54, 63} taken mod 24.

72 is the first number which is divisible by 24, so corresponds to getting back to pin 0

As you can see, this is a cyclic sub-group of our big group, of order 8, generated by '9'.

The size of the subgroup is nothing but our 'k', the number of sides of the shape.

You can also see that k.hcf(a,n) = n

So 'k', the size of the cyclic subgroup of nails is always a factor of 'n'

You can see this in the table above, the sizes of the shapes are 1, 2, 3, 4, 6, 8, 12 or 24 - all the factors of n.

Going the other way, if you have a factor 'k' of 'n' then there is a cyclic subgroup with order 'k'.

Simply take a = n/k

For example, if n = 24 and I want a subgroup of size 8, I could choose a = 3.

That gives the subgroup {0, 3, 6, 9, 12, 15, 18, 21} with size 8, the polygon shown below.

We've seen that:

'3' generates the subgroup {0, 3, 6, 9, 12, 15, 18, 21} of order 8

'9' generates the subgroup {0, 9, 18, 3, 12, 21, 6, 15} of order 8

These are the same numbers in a different order! The two subgroups are the same, but it has two different generators.

That's two generators, how many are there in all?

Well, that's the same as the number of 'a' for which

24     hcf(a,24)

= 8

ie. hcf(a, 24) = 3

'a' must be a multiple of 3, so a = 3c

And it must share no more common factors with 24, so 'c' must share no common factors with 8

The numbers less than 8 which have no common factors with 8 are {1, 3, 5, 7}

So the generators are 3 × 1, 3 × 3, 3 × 5, and 3 × 7.

ie. 3, 9, 15 and 21.

we've seen a = 3 and 9 - here's pictures of a = 15 and a = 21

15 is nothing but -9 (ie. 9 backwards) and 21 is -3 (ie. 3 backwards).

We've got 4 generators, since there are 4 numbers less than 8 and co-prime to 8.

We write this as φ(8) = 4, and this is called Euler's phi function after the Swiss mathematician Leonhard Euler.

In summary, For each factor 'k' of 'n' there is exactly one cyclic subgroup of order 'k'. a = n/k is one generator The others are of the form la where 'l' is one of the φ(k) numbers less than and co-prime to k

Notice also that if the number of nails is prime, n = p, then the factors are just {1, p}.

So if there is a prime number of nails, the string will always go through none, or all nails.