Here's the problem:
There's a recursive tree of whole numbers which can be represented by a function G(n). The number tree looks like this:
following the basic pattern (remember, it's recursive) which looks like this:
where (b) is the second iteration of the tree. (You put a G-tree at every place where it says G on the tree you currently have to get subsequent iterations.) Bored yet? But wait! There's an equation, G(n), to go with all this!
G(n) = n - G(G(n-1)) for n>0
G(0) = 0
If you input a number from a node on the tree as n in the equation, you get its immediate predecessor for G(n). So G(9)=6, etc. And this describes the G-tree.
Here's the dumb hateful tricky part:
Draw the numbered tree the same way, but number the nodes from right to left instead of from left to right.
Now find the function G'(n) (which describes the new "flip-tree").
OMFG I CAN'T.
That's all. I need to sleep now.
First of all ROFL you're on page 137. Secondly, fantastic! Now you get to help me learn math. I don't understand the pictures. Let's start at the very leftmost branch of the first picture. The one that says 14, 15, and 9. That node does not look like (a) in diagram 2. Unless where it says G in (a) in figure 2 is equivalent to the number 14 in figure 1? And if they are equivalent, why are they drawn differently?
ReplyDeleteAh, your issue is just a display convention - the tree is drawn up to a single level so there isn't a choppy top of the tree. You know from diagram 2 that diagram 1 extends upward in that pattern forever, so to make it look pretty they draw it with an even row at the top instead of the "complete" G segments -- which of course can never be complete because they're infinitely recursive and extend forever.
DeleteBasically, the first two diagrams are ways to represent similar things... they're not equivalent, per se. You could take diagram 2 and define G in any way you wanted - like, say it's a copy of the Mona Lisa - and then the nodes on the tree are defined by da Vinci's great work, not by numbers. Diagram 2 describes a recursive /pattern/, whereas diagram 1 is a specifically defined instance of that pattern.
If it's still confusing, do you understand how (a) in diagram 2 is turned into (b), and can you create your own (c) from that? (Hint: ignore diagram 1!)
P.S. myRomeposthasmoreviewsthananyotherpageonthisblogsothere.
ReplyDeleteThis post forced me to do math stuff. I disapprove.
ReplyDeleteI went over to Jon's the other day and he was working it out. Then I had to do it too.
I LOVE EVERYONE AND PUZZLES. But I am still stumped on this. Did he work it out?
DeleteYou may want to check out
Deletehttp://oeis.org/A123070
Still working on it, but I'm going to continue reading GEB instead of goofing off with Bayes Theorem and HPMOR.
ReplyDelete