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Posers and Puzzles

Posers and Puzzles

  1. Donation Acolyte
    Now With Added BA
    30 Oct '02 23:18
    When I posted the first one I meant to post a slightly different problem. Here it is:

    Draw a circle. Draw n points on the circumference. Join up the points with straight lines.
    How many regions can you divide the circle into?
  2. Donation Acolyte
    Now With Added BA
    02 Nov '02 16:18
  3. 03 Nov '02 11:09
    this would be the same problem as the other one, except that the
    number of lines has to be calculated first: the combination of n
    elements in groups of two: n x (n-1) / 2

    And we substitute that into the formula of the other problem

    number of regions = 1 + [N x (N+1)] / 2 (the other problem)
    with N= n x (n-1) / 2
    gives : 1 + [ n x (n-1) x ( n² - n + 2)] / 8

    lets try with
    2 points: 1 line, 2 regions
    3 points: 3 lines 7 regions
    4 points: 6 lines, 22 regions
    etc.....

    or am I talking nonsens (again)?

    sin.
  4. Donation Acolyte
    Now With Added BA
    03 Nov '02 16:06
    Your first answer was correct, but this answer isn't, I'm afraid. You can't make 7 regions with
    just 3 points. Hint: the lines aren't arbitrary, as they were in the first problem.
  5. 03 Nov '02 16:19
    you are right (nothing to be afraid of, LOL). I was a bit naive. There
    are two important differences with the first problem:

    1) In the first problem I could imagine the circle outside ALL
    intersections. So the area's between intersections laying outside the
    circle have to be deducted.

    2) in the first problem, all lines could have intersected at different
    points; Here there are a lot of coinciding intersections.

    Too complicated to answer just like that (at least starting from this
    angle). Back to the study .... Anyone else?
  6. Donation Acolyte
    Now With Added BA
    03 Nov '02 17:15
    I find the easiest way to think about this is by induction. Put a few points down, and see
    what hapens when you add one more point.
  7. Donation richjohnson
    TANSTAAFL
    04 Nov '02 21:02
    2 points = 2 regions.
    3 points = 4 regions.
    4 points = 8 regions.
    5 points = 16 regions.
    6 points = 30 regions?? Am I counting wrong?
  8. Donation Acolyte
    Now With Added BA
    04 Nov '02 23:54
    For 6 I can make more than 30. Don't put the points in a regular hexagon.
  9. Donation richjohnson
    TANSTAAFL
    05 Nov '02 00:09
    with irregular placement I count 31 regions - still not the expected 32.
  10. Donation Acolyte
    Now With Added BA
    05 Nov '02 14:10
  11. 24 Dec '02 10:40
    for n is odd, the terms follow the pattern
    n,n+1,3n+1,5n+1 etc (because you always have that irritating central region)
    and for n is even the terms follow the pattern
    n, 2n, 4n, 6n, 8n etc.
    its a compound series, but thats as far as my memory will get me. i tried to express it in terms of n but i get all fuzzy and irritated. there must be a mathematician or two on this site who see these things when they watch the sky...