This week, I learnt how to solve the problem involving big-Oh with polynomials. Later on this entry, I will show the second question approaches using Polya's method. (note: my first problem solving using Polya's method was in week 7 entry, but soon I realized that instead of solving just the claim, I had to use the problem solving episode, so I am making another problem solving episode using the Polya's method.) The main thing I learnt this week was how to prove big-Oh when the claim is in polynomials. Sadly, with term test #2 and technical issues on Friday, the class was not able to properly cover the part of big-Oh with limits.
Anyway, what I really found challenging this week was when polynomial consists of subtraction and maximum slicing, I had to think really hard to prove that it is part of big-Oh. Moreover, in this week, I had to finish my assignment 2, so I was worried that I wouldn't be able to review this week's course materials. Thankfully, with enough practices and reviews, I was able to understand this week's materials as well.
In addition to that, for big-Oh polynomial cases, I had to understand how I can reinterpret the meaning and thus find a constant C, to prove the special cases.
PROBLEM SOLVING EPISODE USING POLYA'S METHOD
Now here is the problem solving episode approach using Polya's method. My problem solving episode will be on Paper folding which I did in week 3 with Danny Heap. Problem was as follows in the below:
You are given one strip of paper with each hand held by your hand. Then you would need to do fold such that fold would be folded from left to right. (p.s. I was trying to upload the video to explain this more accurately, but my video quality was so bad due to the darkness, so I will just explain it in a word.) For instance, you need to fold the strip of paper so that the left end of the strip is now on top of the right end of strip of the paper. By repeating this steps several times, you need to be able to predict the sequence of ups and downs for any number of fold you did on the strip of the paper. Also I add an additional question that what is the sequence pattern when you did 6 folds without actually folding the strip of paper?
1. Understanding the Problem
First, I know that as I fold my strip of paper as instructed, then the number of ups and downs will rise. Condition given in this problem solving episode is when one fold n times, how will the sequence pattern of ups and downs be represented. Data was obtained through the experiment. My group and I followed the instruction and did a folding from 0 to 4, and here is sequence pattern we got for each folding time. Also note that d represents down creases, u represent up crease, and n represents number which is natural number. Also I need to find the sequence pattern when I did 6 folds without actually folding the strip of paper.
when 0 fold: no up and down can be observed.
when 1 fold: d
when 2 folds: ddu
when 3 folds: ddudduu
when 4 folds: ddudduuddduuduu
2. Devising the Plan
Looking at the data my group obtained, we found the connection that as long as the number of fold is greater than or equal to 1, as when n increases by 1, the number of down appeared on the strip of paper increased twice. Mathematically number of d is 2^(n-1), except when number of fold is 0. As for number of up, the number of up appeared on the paper can be represented as (number of d -1). Now, I will represent this using the table 1.
Table 1
Number of fold
|
Sequence Pattern
|
Number of u
|
Number of d
|
0
|
none
|
0
|
0
|
1
|
d
|
0
|
1
|
2
|
ddu
|
1
|
2
|
3
|
ddudduu
|
3
|
4
|
4
|
ddudduuddduuduu
|
7
|
8
|
Soon our group realized there was another relationship between number of fold and the sequence pattern using the idea of the symmetry. So I am going to rewrite my sequence pattern using the table 2.
Note
d: first symmetry, d or u: second symmetry, d or u: third symmetry, d or u: fourth symmetry
Table 2
Number of fold
|
Pattern
|
Number of different Colour Symmetry
|
0
|
No pattern observed
|
0
|
1
|
d
|
1
|
2
|
d d u
|
2
|
3
|
d d u d d u u
|
3
|
4
|
ddudduuddduuduu
|
4
|
(p.s. please the pattern you might see on the blog may be not organized. I am deeply sorry for the inconvenience)
Judging from the table 2 above, I was able to find out that for n folds, there are n different numbers of symmetry which can be observed on the sequence pattern. Moreover, as the number of n increases by 1, on the left side of (n-1)th symmetry, there is addition of d, and on the right side of (n-1)th symmetry, there is an addition of u. For instance, when n = 3, on the second symmetry I need to add d on the left of
d and u, then add u on the right side of d and u.
Also I found that the very first symmetry can only be
d.
3. Carrying out the Plan
Using what I know from section 2.Devising the plan, now I will be able to find the sequence pattern when number of fold(n) is 6. First I know that there will be 6 different colour symmetry. Also I know that there are 32 d in the sequence pattern through use of 2^(n-1) = 2^(6-1) = 2^5 = 32. As for the number of u appeared on the sequence will be then 32 -1 = 31. Also from section 2, I know that there is one more d from the fact that there is only one d observed when there is 1 fold on the strip of the paper.
note that
d or
u: fifth symmetry, d or u: sixth symmetry
Thus, when you did 6 folds, the sequence of pattern is as below:
ddudduuddduuduudddudduuudduuduudddudduuddduuduuuddudduuudduuduu
Also when you did n folds:
-you need to add d on the left side of (n-1)th symmetry from the previous pattern observe which is when there is n-1 folds.
-you need to add u on the right side of (n-1)th symmetry from the previous pattern observe which is when there is n-1 folds.
4. Looking back
To check whether the sequence pattern when n=6 is correct, I redid the problem and also counted the number of u and d to ensure that when there is 6 folds, there are 31 u and 32 d appear in the sequence. I can also use my solution to the n folds on the strip of paper to find the other sequence pattern such as when n=7,8,9...etc. Finally the solution to this problem can be easily find when you arrange the sequence pattern you observed in the middle, and carefully examine one by one to find its relationship. I also believe that this problem solving episode through Polya's method enhanced me how to approach to different mathematical problems.