This drawing scheme will visit every vertex in the polygon before repeating if and only if the number of vertices N and the number K is relatively prime (i.e. have no common factors). I'll leave the proof to you. In the previous example, N = 5 and K = 2 are relatively prime so that drawing method visits every point before repeating.

Note that K = 1 is relatively prime to any number N so drawing with K = 1 visits every vertex before repeating. That just gives the polygon itself.

Also note that K = k and K = N - k draw the same "star." Try K = 2 and K = 3 for a pentagon.

The picCanvas control's Paint event handler draws the stars. It allocates space for the polygon's points and places them evenly spaced around a circle. It then loops over the numbers 1 through N \ 2 and tries using them as K. It doesn't need to examine values N \ 2 through N because of the second note above.

For each skip (K) value, the program determines the greatest common divisor (GCD) of K and N. If the GCD is 1, then N and K are relatively prime so the program calls subroutine DrawStar to draw the "star" and draws a label showing the number K.

' Draw the applicable stars.
Private Sub picCanvas_Paint(ByVal sender As Object, ByVal e _
    As System.Windows.Forms.PaintEventArgs) Handles _
    picCanvas.Paint
    If m_NumPoints < 3 Then Exit Sub

    Const RADIUS As Integer = 40
    Dim x, y As Integer

    ' Position the original points.
    Dim pts(m_NumPoints - 1) As PointF
    Dim theta As Double = -PI / 2
    Dim dtheta As Double = 2 * PI / m_NumPoints
    For i As Integer = 0 To m_NumPoints - 1
        pts(i).X = RADIUS * Cos(theta)
        pts(i).Y = RADIUS * Sin(theta)
        theta += dtheta
    Next i

    ' Draw stars.
    y = RADIUS + 5
    x = RADIUS + 5
    Dim sf As New StringFormat
    sf.Alignment = StringAlignment.Center
    sf.LineAlignment = StringAlignment.Near
    e.Graphics.Clear(picCanvas.BackColor)
    For skip As Integer = 1 To m_NumPoints \ 2
        ' See if they are relatively prime.
        If GCD(skip, m_NumPoints) = 1 Then
            ' Draw the star.
            DrawStar(e.Graphics, x, y, pts, skip)

            ' Draw a label.
            e.Graphics.DrawString(skip.ToString, _
                picCanvas.Font, Brushes.Black, x, y + _
                RADIUS + 2, sf)

            x += 2 * RADIUS + 5
            If x + RADIUS >= picCanvas.Width Then
                y += 2 * RADIUS + 15
                x = RADIUS + 5
            End If
        End If
    Next skip
End Sub

' Draw a star centered at (x, y) using this skip value.
Private Sub DrawStar(ByVal gr As Graphics, ByVal x As _
    Integer, ByVal y As Integer, ByVal orig_pts() As _
    PointF, ByVal skip As Integer)
    ' Make a PointF array with the points in the proper
    ' order.
    Dim pts(m_NumPoints - 1) As PointF
    For i As Integer = 0 To m_NumPoints - 1
        pts(i) = orig_pts((i * skip) Mod m_NumPoints)
    Next i

    ' Draw the star.
    gr.TranslateTransform(x, y)
    gr.DrawPolygon(Pens.Blue, pts)
    gr.ResetTransform()
End Sub

' Return the greatest common divisor (GCD) of a and b.
Public Function GCD(ByVal a As Long, ByVal b As Long) As _
    Long
    Dim tmp As Long
    Dim remainder As Long

    ' Make a >= b.
    a = Abs(a)
    b = Abs(b)
    If a < b Then
        tmp = a
        a = b
        b = tmp
    End If

    ' Pull out remainders.
    Do
        remainder = a Mod b
        If remainder = 0 Then Exit Do
        a = b
        b = remainder
    Loop

    Return b
End Function