1. Forum
  2. Usenet
  3. REC.PUZZLES

  • How Many Regions?

    From David Entwistle@21:1/5 to All on Sun Jul 27 11:42:35 2025
    A curiosity from the Book of Numbers.

    If you place n dots, irregularly, on the perimeter of a circle, how many separate regions are formed within the circle when all dots are joined in
    all possible ways? The dots should be placed such that only two lines
    intersect at any one point.

    The sequence starts:

    1 dot, 1 region.
    2 dots, 2 regions.
    3 dots, 4 regions.
    ...

    Can you extend the sequence to 6 dots?

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Ilan Mayer@21:1/5 to All on Sun Jul 27 13:27:10 2025
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

    A curiosity from the Book of Numbers.

    If you place n dots, irregularly, on the perimeter of a circle, how many separate regions are formed within the circle when all dots are joined in
    all possible ways? The dots should be placed such that only two lines intersect at any one point.

    The sequence starts:

    1 dot, 1 region.
    2 dots, 2 regions.
    3 dots, 4 regions.
    ...

    Can you extend the sequence to 6 dots?


    See https://oeis.org/A000127

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Heathfield@21:1/5 to David Entwistle on Sun Jul 27 15:51:17 2025
    On 27/07/2025 12:42, David Entwistle wrote:
    A curiosity from the Book of Numbers.

    If you place n dots, irregularly, on the perimeter of a circle, how many separate regions are formed within the circle when all dots are joined in
    all possible ways? The dots should be placed such that only two lines intersect at any one point.

    The sequence starts:

    1 dot, 1 region.
    2 dots, 2 regions.
    3 dots, 4 regions.
    ...

    Can you extend the sequence to 6 dots?

    This is a well-known problem, so rather than spoil it for you I
    hope you'll take my word for it when I say "yes".

    --
    Richard Heathfield
    Email: rjh at cpax dot org dot uk
    "Usenet is a strange place" - dmr 29 July 1999
    Sig line 4 vacant - apply within

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to David Entwistle on Mon Jul 28 13:02:30 2025
    On Sun, 27 Jul 2025 11:42:35 -0000 (UTC), David Entwistle wrote:

    Can you extend the sequence to 6 dots?

    SPOILER.
    POILER.
    OILER.
    ILER.
    LER.
    ER.
    R.
    .

    Having spent a couple of hours refreshing my algebra skills, I think the candidate polynomial would be:

    f(n) = (n^4)/24 - (n^3)/4 + 23*(n^2)/24 - 3*n/4 + 1

    That is based on the finite differences of the sequence for 0 <= n <= 10.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Heathfield@21:1/5 to David Entwistle on Tue Jul 29 06:32:35 2025
    On 28/07/2025 14:02, David Entwistle wrote:
    On Sun, 27 Jul 2025 11:42:35 -0000 (UTC), David Entwistle wrote:

    Can you extend the sequence to 6 dots?

    SPOILER.
    POILER.
    OILER.
    ILER.
    LER.
    ER.
    R.
    .

    Having spent a couple of hours refreshing my algebra skills, I think the candidate polynomial would be:

    f(n) = (n^4)/24 - (n^3)/4 + 23*(n^2)/24 - 3*n/4 + 1

    You can check yourself against Wikipedia...

    https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas

    ... or OEIS:

    https://oeis.org/A000127

    --
    Richard Heathfield
    Email: rjh at cpax dot org dot uk
    "Usenet is a strange place" - dmr 29 July 1999
    Sig line 4 vacant - apply within

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Charlie Roberts on Tue Jul 29 07:19:18 2025
    On Sun, 27 Jul 2025 14:40:40 -0400, Charlie Roberts wrote:

    Hi Charlie.

    If n points are placed in a circle and all of them are connected,
    what is the number of intersection points made by the diagonals?


    Could you clarify the above, or provide a pointer to the question? Is
    that, n points are placed on the perimeter of a circle, or n points are
    placed somewhere within a circle? If the second, are we extending straight lines through the points to the point they intersect with the circle,
    rather than having the line end at the points? When you say diagonals, I'm thinking chords?

    Thanks.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Richard Heathfield on Tue Jul 29 07:29:00 2025
    On Tue, 29 Jul 2025 06:32:35 +0100, Richard Heathfield wrote:

    You can check yourself against Wikipedia...

    Thanks. It's nice to be able to derive a result from a problem. Having got
    the wrong answer four times before getting the right answer, I think my
    algebra may have improved with practice. My level of mistakes certainly reduced.

    I'm not sure I have the deep understanding of 'why', though.

    I had been looking at the lazy caterer's sequence and derived the
    polynomial for that. The How Many Regions problem gave me an uneasy
    feeling I may have missed something there.

    https://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence

    All good fun though and useful experience.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Charlie Roberts on Wed Jul 30 08:19:51 2025
    On Tue, 29 Jul 2025 09:20:19 -0400, Charlie Roberts wrote:

    2) the vertices of a regular polygon -- which is treated in the paper I mentioned, or

    The number of intersections of chords from the vertices of the regular polygons, with an odd number of vertices, looks to be more my level of
    problem. That'll limit the number of intersecting chords to a maximum of
    two. The polygons with even vertices 'looks' hard.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Charlie Roberts on Thu Jul 31 13:24:34 2025
    On Wed, 30 Jul 2025 09:23:08 -0400, Charlie Roberts wrote:

    Yes, at least to me. I was amazed when I saw the expression for the
    number of intersections and regions for the case of regular polygons.

    For the polygons with an odd number of vertices 1, 3, 5, 7, 9, 11, 13 etc.
    the polynomial describing the number of intersections looks to be:

    f(n) = (n^4)/24 - (n^3)/4 + 11 * (n^2)/24 - n/4.

    The complications associated with the intersections of more than two
    diagonals, at the same point, for the polygons with an even number of
    vertices, probably takes the problem outside the realm of my abilities.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From David Entwistle@21:1/5 to Charlie Roberts on Wed Jul 30 07:04:12 2025
    On Tue, 29 Jul 2025 09:20:19 -0400, Charlie Roberts wrote:

    Sorry, I should have been clearer. What I meant was a generalsation of
    the problem you posed. The n points are all on the circumference of the circle just as in your case. But, no condition is made on their
    (relative) positions.

    Thanks. I understand now.

    --
    David Entwistle

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From James Dow Allen@21:1/5 to All on Sun Aug 3 12:57:06 2025
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

    A curiosity from the Book of Numbers.

    If you place n dots, irregularly, on the perimeter of a circle, how many separate regions are formed within the circle when all dots are joined in
    all possible ways? The dots should be placed such that only two lines intersect at any one point.

    I'm not sure but isn't one way of specifying the restriction to say that the points are "in general position"?

    This problem clicks my nostalgia button. My excellent high school math teacher, seeing I needed a challenge, assigned me this problem; I developed difference equations to get a quartic solution; and went back to day-dreaming.

    I'm five times older now, but, in some ways, thrice as dumb.

    James

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)