# 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 ðŸ™‚

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?

Tags

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 ðŸ™‚

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!!!

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?

>> 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…