Recent content by Puzzles

I actually managed to find an awesome proof for the first problem at slide 3 of http://www.math.ucsd.edu/~gptesler/184a/slides/184a_trees_17-handout.pdf Now I've only to solve the second problem, if anyone has any hints, it would be grand. :)

Hi! I'm struggling with these two problems: 1. If for whichever two vertices a and b in the graph G there is only one simple path from a to b, then the graph is a tree. Eh... isn't this part of the definition for a tree? I really don't even know where to start with proving this statements...

Yeah, I'm a little popular in my uni for screwing up at least a quarter points of each exam by making silly mistakes like that. :D an = 1/sqrt(5)(1+sqrt(5))n - 1/sqrt(5)(1-sqrt(5))nTo the follow up question, I get the same result... :( My gut tells me I'm just solving it wrong, as usual. The...

Well, a1 = 2 = c1(1+sqrt(5)) + c2(1-sqrt(5)) a2 = 4 = c1(1+sqrt(5))2 + c2(1-sqrt(5))2 c1 + c1sqrt(5) + c21 - c2sqrt(5) = 2 6c1 + 2c1sqrt(5) + 6c21 - 2c2sqrt(5) = 4 When I multiply the first equation by 2, and subtract it from the second, I get c1 = c2 And, replacing that in the first...

Oh thanks, that makes a lot of sense. And yeah, I just wrote it down directly because I got c1 = c2 = 1.

How did you come to that recurrence? For the roots, I get 1+sqrt(5) and 1-sqrt(5). (I still can't figure out how to format it properly, I'm sorry :( I keep trying to) Based on this, I get: an = (1+sqrt(5))n + (1-sqrt(5))n

Update: I had a talk with the professor today, and turns out there was a typo in the assignment - the board is supposed to be 2xn, not 1xn as it was written. She also confirmed we're considering 1x2 sized dominoes. I've found several solutions online, which look pretty much the same like the...

Yeah, exactly! My approach is never mathematical enough. Thanks a bunch, again!

Oh, right, thanks! Isn't it simply ak = 2k/2?

I'm sorry, I'm not sure I know what "closed form" refers to.

ak = 2ak-1. Same as ak = 2n/2, which doesn't rely on the previous solution. I have trouble trusting this solution...

I agree that it's an awfully constructed problem, honestly I'm not even sure I understand it. Somehow I got to solve it in the next few hours. I've been thinking about it a lot. Let's say one domino is blue, the other is green. 1 x 1: 0 1 x 2: 2 different ways, we can place the blue one, or...

Hey everyone, it's me again with yet another recurrence equation I've been stuck with: Using recurrence relations (recurrence equations... is it the same thing?), solve the following: There is a chart with dimensions 1xn. We have dominoes in two different colors which we should use to fill up...

Thank you very, very much, and I'm sorry for being useless throughout this.

k1 = 3/4 k2​ = 17/4 Do I just replace these in the homogeneous solution and get the final solution?