Posers and Puzzles

Posers and Puzzles

  1. Standard memberroyalchicken
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    11 Feb '03 22:08
    ...in under a week.

    Take one dot. From there, you can make a 3-dot equilateral triangle, a four-dot square, a five dot regular pentagon, etc. From those, you can build a 6-dot trianlge, a 9-dot square, a 12-dot pentagon, etc. So any number that is the total number of dots in a regular r-gon so built is a r-gonal number. For convenience, say 0 is an r-gonal number for any r. Can someone prove that EVERY positive whole number can be represented in at least one way as the sum of r r-gonal numbers, for every r?

    Hints as needed, although I bet someone can come up with a prettier proof than mine...
  2. Standard memberzach918
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    11 Feb '03 22:21
    MARK-
    interesting puzzle....hate to change the subject but, why did you delete your profile pic 😕 it was a good picture 🙂
  3. Standard memberroyalchicken
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    11 Feb '03 22:25
    I never had a profile pic-I haven't got a star.
  4. Standard memberthire
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    11 Feb '03 23:141 edit
    Originally posted by royalchicken
    I never had a profile pic-I haven't got a star.
    I guess he knows... this 😉
  5. Steelers Country
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    11 Feb '03 23:54
    Originally posted by zach918
    MARK-
    interesting puzzle....hate to change the subject but, why did you delete your profile pic 😕 it was a good picture 🙂
    are you talking about the T1000 Mark maby?
  6. Standard memberroyalchicken
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    12 Feb '03 18:51
    Probably. But enough of this! Anyone solve it yet? Or should we wait for Acolyte?
  7. Donationmaggoteer
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    13 Feb '03 07:54
    Originally posted by royalchicken
    ...in under a week.

    Take one dot. From there, you can make a 3-dot equilateral triangle, a four-dot square, a five dot regular pentagon, etc. From those, you can build a 6-dot trianlge, a 9-dot square, a 12-dot pentagon, etc. So any number that is the total number of dots in a regular r-gon so built is a r-gonal number. For convenience, say 0 is ...[text shortened]... ery r?

    Hints as needed, although I bet someone can come up with a prettier proof than mine...
    This isn't anywhere near the problem, but first things first, I want to make sure we are thinking of the same r-gonal numbers before I start racking my brain. So, if we are thinking of the same number, then the general formula for the number of points in a given r-gonal number is:
    N[r,w] = (4 + r(w - 1) - 2w)w/2
    where r is the number of sides and w is the number of dots for a given side, vertices included. r>=3, w >=2.
    Am I ok so far? Hate to start thinking about the proof if I'm not thinking of the same thing as you!
  8. DonationAcolyte
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    13 Feb '03 09:41
    Originally posted by maggoteer
    This isn't anywhere near the problem, but first things first, I want to make sure we are thinking of the same r-gonal numbers before I start racking my brain. So, if we are thinking of the same number, then the general formula for the number of points in a given r-gonal number is:
    N[r,w] = (4 + r(w - 1) - 2w)w/2
    where r is the number of sides and w is t ...[text shortened]... ok so far? Hate to start thinking about the proof if I'm not thinking of the same thing as you!
    I still don't see how pentagonal and higher numbers work graphically 😳
  9. Donationmaggoteer
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    13 Feb '03 19:101 edit
    Originally posted by Acolyte
    I still don't see how pentagonal and higher numbers work graphically 😳
    Assume I'm thinking of the same shape as royalchicken. Then for a given polygonal shape, say a pentagram, imagine as follows. A next bigger pentagram, extends the edge length of the next smaller pentagram by one dot; but they always share the two edges adjacent to the single "common" vertice. Notice that along those two shared edges, the dots that were vertices of the smaller pentagram are now "edge" vertices of the bigger pentagram.

    Um, bejeese,I just read that. I doubt it helped! Yeah, this is a time when a picture would be worth about a mole of words....
    And I probably should have recast my formula; as I've written it, I consider any r-gonal number of length 1 to be a single dot. The first square, ie the 4 dot one, is in my formula P[4,2]. P[4,1] is the "degenerate" square with one dot. 😕
  10. Standard memberroyalchicken
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    14 Feb '03 02:262 edits
    Acolyte & Maggoteer (quite possibly the two coolest usernames)-

    I'll tell you my general formula for clarification. It barely helps at all for the proof. Let A(r,n) be the nth r-gonal number. Then:

    A(r,n) = n + (r-2)(n-1)n/2 = n + (r-2)A(3,n-1)

    So some of the the 2-gons, 3-gons, 4-gons, 5-gons are:

    0, 1, 2, 3, 4, 5....

    0, 1, 3, 6, 10, 15...

    0, 1, 4, 9, 16, 25...

    0, 1, 5, 12, 22, 35...

    Basically, you will recognise that each term in one of these sequences is formed by adding the next term in an arithmetic series with common diffrence r-2 and initial term 1 (except the 0's). I want you to prove that every natural number is the sum of 3 3-gons, 4 4-gons, etc.

    My proof took two weeks of evenings and is basically analytic in character. Something combinatorial might be more natural. Good luck.


    ~mark (Je ne suis pas un poulet, mais je pense que je me conduis comme un poulet quand je joue un match de la chesse.)

    Sorry, maggoteer, I think I explained it poorly the first time. When I discussed these numbers in terms of polygons, this was mostly because I was following Acolyte's advice and not making this look like a math(s) exam. Think of it, if you like, in the way I expressed above.
  11. Standard memberroyalchicken
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    14 Feb '03 02:29
    Originally posted by Acolyte
    I still don't see how pentagonal and higher numbers work graphically 😳
    It's okay. Visualisation can be a handicap.

    Just build one more layer on the outside. THe total number for pentagons is n(3n-1)/2 => 0,1,5,12...
  12. Donationmaggoteer
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    14 Feb '03 04:341 edit
    Originally posted by royalchicken
    Let A(r,n) be the nth r-gonal number. Then:

    A(r,n) = n + (r-2)(n-1)n/2 = n + (r-2)A(3,n-1)
    At least our formula expand out to the same! So I'm at leas thinking what your thinking. Um..not sure I'll get much further....😕 Acoylte, the field is yours!😵🙄
  13. Donationbbarr
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    14 Feb '03 05:21
    Originally posted by royalchicken
    Acolyte & Maggoteer (quite possibly the two coolest usernames)-

    I'll tell you my general formula for clarification. It barely helps at all for the proof. Let A(r,n) be the nth r-gonal number. Then:

    A(r,n) = n + (r-2)(n-1)n/2 = n + (r-2)A(3,n-1)

    So some of the the 2-gons, 3-gons, 4-gons, 5-gons are:

    0, 1, 2, 3, 4, 5....

    0, 1, 3, 6, 10, ...[text shortened]... t making this look like a math(s) exam. Think of it, if you like, in the way I expressed above.
    So did you construct a proof by mathematical induction, using the assumption that 0 is an r-gonal number for any r as its base?
  14. Standard memberroyalchicken
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    14 Feb '03 16:441 edit
    Originally posted by bbarr
    So did you construct a proof by mathematical induction, using the assumption that 0 is an r-gonal number for any r as its base?
    Hey! Slyly trying to get a hint! But it is Valentine's day...Tried that. That was, it seems, the most reasonable thing to do. But I was unsuccessful. So I let k(r,n) be the nth number that satisfies the conjecture. I then let X(r,x) be the number of k(r,n) between 0 and x inclusive for some real number x. All I showed was that x = X(r,x) + O(1) for all x in R. Obviously, this implies that since X is a step function, the set of k(r,n) is equal to N for all r. But I won't go into any more detail, except under torture.

    The use of 0 was just so that exactly r r-gons are used. For example, 6 = 3+3+0 or 6+0+0, but it cannot be formed as the sum of three 3-gons in any other way.
  15. DonationAcolyte
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    14 Feb '03 17:58
    Originally posted by royalchicken
    The use of 0 was just so that exactly r r-gons are used. For example, 6 = 3+3+0 or 6+0+0, but it cannot be formed as the sum of three 3-gons in any other way.
    Or equivalently, any number can be expressed as the sum of at most r r-gonal numbers for all r.
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