We received a number of good solutions to this problem.

Tabs from St. Mary's School noticed that:

If the number of points is a prime number you will hit all the points every time.If the number of points is not prime, you will hit all the points when the number of steps does not share any factors.

Eg, for 10 points you will hit every one with 1 step, 3 steps, 7 steps and 9 steps but you won't hit all the points with 2 steps, 4 steps, 5 steps, 6 steps or 8 steps as these numbers all share a common factor with 10.

Esther Shindler made similar observations:

If the number of points is a prime number, then any number of
steps will hit all points.

e.g. 11 points are always hit whichever step size you
use.

For other numbers we have to find their prime factors.

If the step size is one of these, or a multiple of any of
them, it will not hit all points. Other step sizes will.

For example, since 14 = 7 x 2, for 14 points,

the step sizes that will not hit all the points are 2, 4, 6,
7, 8, 10, 12,

and the step sizes that will hit all points are 1, 3, 5, 9,
11, 13.

Pippa and Sophie from The Mount School in York produced a table of results and added some comments

You can work out which step sizes will hit all the points for
any number of points by

a) Finding the factors of the
number.

b) Write down all the numbers that
aren't factors or multiples of factors.

c) Write down the number 1 in the
list of step sizes.

For prime numbers, such as the number 5, all possible step sizes will work. All the number of step sizes are even.

If the number of points is prime the formula for the number of
step sizes is n-1 where n is the number of points. All the step
sizes are even.

Yanqing from Devonport High School for Girls produced a useful summary of her results:

When the circle has 8 dots, I found that you can hit all the points with steps of 1, 3, 5 and 7 points. Step sizes of 2, 4 and 6 misses some points.

When the circle has 9 dots, step sizes of 1, 2, 4, 5, 7 and 8 will hit all the points, and 3 and 6 misses some points.

In a circle of 10 dots, you can hit all the points with step sizes of 1, 3, 7 and 9. 2, 4, 5, 6 and 8 misses some points.

I found a pattern in these results. All the step sizes that misses some points share a factor other than 1 with the circle size. From these, I have concluded that as if the highest common factor between the number of dots on the circle and the step size is 1, you can hit all the points.

With a circle of 5 dots, you can hit all the points with any number of steps. Other circles that do this are 3, 7, 11, 13, 17 etc. These are all prime numbers. This fits the pattern: prime numbers do not have any factors larger than 1, and so all the step sizes can hit all the points.

The number of steps that will ensure that all points are hit are the same as the number of points in the circle. This is because you will need to go to all the points, and this would mean you would need a step for each point.

Stephen from Pike County High School came to a similar conclusion:

To determine if a step size will hit all points of a circle,
find the Greatest Common Factor (GCF) between the number of points
and the step size. If the GCF is 1, then the step size will hit all
points of a circle.

For example, let's look at a circle with ten points.

One will work. GCF is 1.

Two will not work. GCF is 2.

Three will work. GCF is 1.

Four will not work. GCF is 2.

Five will not work. GCF is 5.

Six will not work. GCF is 2.

Seven will work. GCF is 1.

Eight will not work. GCF is 2.

Nine will work. GCF is 1.

This gives us four step sizes that will hit all points in a
ten point circle.

This works with all circles, regardless of the number of
points.

Well done to you all.