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Puzzle: a little geometry problem, and a sequence question

  • Article

This is not really a puzzle, but a real geometry problem. Let's take a random triangle (ABC), and let's assume that the angle bisector from A intersects BC in the point D. Proof that:

   AD ^ 2 = AB * AC - BD * CD

Here is the figure, drawn in MSPAINT.EXE as you can see :-)

And now, a real math puzzle. Here is a sequence of sequences of numbers.

     1
1,1
2,1
1,2,1,1
1,1,1,2,2,1
3,1,2,2,1,1
...

What comes next?

Comments

  • Anonymous
    August 05, 2005
    Second question:
    1,3,1,1,2,2,2,1

  • Anonymous
    August 06, 2005
    3,1,2,2,1,1
    1,3,1,1,2,2,2,1 comes next . This is a good one. i think i first saw it on google aptitude test.

  • Anonymous
    August 07, 2005
    1,3,1,1,2,2,2,1
    And then:
    1,1,3,1,2,1,3,2,1,1

    Very nice!!!

  • Anonymous
    August 07, 2005
    Alex got one wrong on his last one.
    3,1,2,2,1,1
    1,3,1,1,2,2,2,1
    1,1,1,3,2,1,3,2,1
    then,
    3,1,1,3,1,2,1,1,1,3,1,2,1,1

  • Anonymous
    August 07, 2005
    John Horton Conway's Sequence
    1,3,1,1,2,2,2,1 (one 3, one 1, two 2, two 1)
    http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A005150

  • Anonymous
    August 08, 2005
    This is a scanned immage of the cover of the issue in which the article of Prof. Conway appeared :-)
    http://www.srcf.ucam.org/archim/eureka/46/cover.jpeg

    Wow, for £1 only, plus postage and packing, you can buy the original issue!

    For curious minds, there is a tremendous relationship of this sequence with the 92 sub-uranium stable elements!!!

  • Anonymous
    August 08, 2005
    The comment has been removed

  • Anonymous
    August 08, 2005
    >> For curious minds, there is a tremendous relationship of this sequence with the 92 sub-uranium stable elements!!!

    Interesting...

    Actually, the heaviest stable element is lead (with the atomic number = 82). Uranium is the heaviest element which is abundant enough to be noticed. Elements with a higher atomic mass are present in nature too, for example various isotopes of plutonium (Pu-239, Pu-238) exist in nature but in extremely low quantities, for example Pu-239 forms by neutron capture in U-238, and neutrons are being constantly emmited in all sorts of conditions. In fact the natural nuclear reactors at Oklo were producing a significant quantity of plutonium (which is gone by now - it decayed long time ago). Also, supernova explosions produce vast quantities of heavy elements (including transuranic ones like Pu, Am, Cm, etc).

  • Anonymous
    August 14, 2005
    The comment has been removed

  • Anonymous
    August 14, 2005
    http://de.wikipedia.org/wiki/Benutzer:MovGP0/Dreieck

  • Anonymous
    August 16, 2005
    I think this applies to the Cosmological Theorem see: http://mathworld.wolfram.com/CosmologicalTheorem.html

  • Anonymous
    August 18, 2005
    The geometry problem from my previous math puzzle has a nice solution. I particularly like it because...

  • Anonymous
    March 24, 2008
    PingBack from http://caferestaurantsblog.info/antimail-puzzle-a-little-geometry-problem-and-a-sequence-question/

  • Anonymous
    April 04, 2008
    PingBack from http://drinksairportsblog.info/antimail-puzzle-a-little-geometry-problem-and-a-sequence-question/