Puzzle: a little geometry problem, and a sequence question


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 (14)

  1. peterchoicm says:

    Second question:

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

  2. vivek garg says:

    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.

  3. Alex says:

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

    And then:

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

    Very nice!!!

  4. ms says:

    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

  5. 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

  6. 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!!!

  7. AdiOltean says:

    Silly me, posting a puzzle from the Google aptitude test. I completely forgot about that…

    By the way, nobody took a crack to the geometry problem?

  8. AdiOltean says:

    >> 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).

  9. MovGP0 says:

    didn’t solved it, but got the proof for AB/BD = AC/CD: <a href="http://de.wikipedia.org/wiki/Benutzer:MovGP0/Dreieck&quot;

    >my notices</a> (<em>note:</em> D = L<sub>a</sub>)<br />

    Maybe I’m thinking to complex. For this weekend I give up.

  10. MovGP0 says:

    I think this applies to the Cosmological Theorem see: http://mathworld.wolfram.com/CosmologicalTheorem.html

  11. Antimail says:

    The geometry problem from my previous math puzzle has a nice solution. I particularly like it because…