Non-equivalent Ternary Tree.

After several weeks' study on induction prove. I gotta a better sense about the ternary tree question than I was first time doing assignment 1. This question is interesting now. I checked some books about the property of trees. I kinda found a closed form formula for this question. which is 3n!/(2n!n!(2n+1)). However, It is kinda not that easy to prove. My way to solve this problem is using Sigma for different trees and get the product of them. It is sound silly comparing to that perfectly closed form formula, but it is easy to proof, and understand. 

Assignment 1 Remark??

I have talked to Danny after term test 1. I am told to send him an email for remark. I did so. However, I did not seen any update from online marking tool or an email back from Danny. Did you received my email? or else?

Term Test 1

This test is fairly easy, just spent 30 minutes to finish it. But still i did not get a perfect mark. The problem is the question 1. The one that require us to prove for n>5, P(n). I use complete induction here. I know I am gonna use all the cases for 0 to n-1. However, maybe I think it is too easy to dig it deep. The base case in my proof is not adequate. Since I will use P(n-2) thus, I have to show the base case for n=8, since n-2=6 this is where the proof should start. My base case is simple P(6), P(7) without P(8). Two marks are taken away, because of the base case problem.

Many time, When I am doing an induction prove question, i am focus on how to show P(n) to P(n+1) without pay much attention on the base case. (In most case, it is too obvious to say any words) However, like the question 1 in the term test, the base case is actually the key for this proof, maybe the only tricky in this question. So, next time, when i am dealing with induction proof, i should take care base case carefully, coz tricks born there.

Busy

I am so busy recently, we have a lots of assignments due last friday, and before that we have 236 term test 1. This week, yesterday, we just finished a statistic test. So there is a lot of missing updates for csc236.

DIY Closed Form

I am a little bit confused about the question in class. H(n) is defined as 

H(n)=0 for n=0

H(n)=2 for n=1

H(n)= 3H(n-1)-2H(n-2) for n>1

I can understand the alpha and beta manipulation staff, but how to get the h0=1, h1=2 and the formula h^2=3h-2? Why it is as this?

What I did for this question is to list down for n from 0 to 6, we can quickly found the patten that H(n) = 2^(n+1)-2, since every H(n) is the power of n+1 and less than 2. But how to get h0 and h1 and what are those symbols stand for?

A1 Schema

Just checked my a1 mark online, it is really bad. I am question about the marking schema? Why my solution only get 2 out of 5. And the reason is just some gaps in proof. Wondering how detail it should be when we are writting our assignment. Why it is keep saying we have gaps in our proof? We did everything correct but only get half mark?? I am very dispointed about the result. Feel a little down about this course, if the TA keep try to mark assignment or any test like this way. By the way, for the question two, Danny replied my slog said my ways may be a good one, but the TA just stick to the way on the sample solution, give me 2 of 5 on this one. And wondering any way to remark on this assignment?

May possible to check my work at g8simon.

Prep Term Test 1

Term Test 1 is settled on the next Friday. I think the main topic on this tt1 is induction prove.

Assignment 1

The assignment 1 sample solution is posted yesterday. I read through this solution today, and the question 1 is almost the same as my solution. It could solved by adding one extra triangle.

The question 2 is slight different from my work, despite, we all divide the n+1_meal menus to the one include n+1_meal(s1) and the one not include n+1_meal.(s2) We know that the n+1_meal that not include the newly n+1_meal partition is identically the same as n_meal. My way does not inverse the s2 menu, instead I just jump back and forth from s1 to s2, which draw in graph is like a 0-1 voltage diagram. Which we has s1[0] to s2[0] to s2[1] to s1[1] to s1[2] to s2[2] to... and so on. I found my work is working but kinda not concise as the solution which is simple inverse the s2 and rest of work is just concatenate the head and tail. I will see whether my work will get full mark or not when the marked paper handed back.

The question 3 is interesting. Actually I have no idea about this question at all when I first read it. The I read some info. like wiki and other websites that have explanation of this kind question. After I grab all the key concept, I write my own work on this question, but it turns out, it is similarly to the one posted as solution. Maybe this is the standard solution or way to solve this kinda question? For the square root of 5 part, I use the same patten as the question to prove square root of 2 is irrational, simply because I can handle that proof well. So I just slight change and parameters, and leave the rest of answer as similar as proving square root of 2 question.