We get a polygon that doesn't intersect itself when the string goes round only once.
ie. when a = hcf(a, n)
This means that a is a factor of a and of n.
ie. a divides into n.
Here's an example, with n = 24 and a = 6
k =
24 hcf(24,6)
=
24 6
= 4
so there are 4 steps - ie. a square.
The number of complete multiples of 'n' is
6 hcf(24,6)
=
6 6
= 1
But hang on, what about something like n = 24 and a = 22?
In this case, since we're stepping around by more than 180 degrees each time, the string goes 2 steps backwards around the circle each time(try it yourself!)
This is equivalent to just stepping around 2 each time, but in reverse. The shape obviously doesn't change.
So we missed this result before, our calculations assumed that in stepping round 22 times we would go the long way round, through zero and pick up an extra revolution.
Actually, we should think of this case as a = 22 - 24 = -2 (2 steps in the other direction).
When n = 24 and a = -2,
k =
24 hcf(-2,24)
=
24 2
= 12
(the hcf of -2 and 24 is 2, I'm taking the largest positive number that divdes exactly into each)
And the number of revolutions is
-2 hcf(24,-2)
=
-2 2
= -1
This is correct, we have a 12 sided shape with -1 revolution around the circle!
So we've seen that for n = 24, we should allow the values of 'a' to be:
String Patterns