Penny Piles

In Friday’s class, we were given an interesting exercise, the question shown in the image below.

Like last time, the purpose for this exercise is to help us familiarize with the procedures of problem solving. I will briefly explain how I solved each of the 3 questions, and reflect on how I could’ve done better.

Q1: 64 pennies to start with, sort it so that one drawer has 48 pennies

A1: The solution to this questions is straightforward. I move the pennies twice to obtain 48 pennies in one drawer.

Q2: Find any numbers in range [0, 64] that are impossible to achieve.

A2: First off, I need to find ways to get the numbers from 0 to 32, since any n above 32 exists if its counterpart 64 – n exists. I approached this problem by drawing a tree diagram. It would be unrealistic to draw out all possibilities on the tree diagram, so instead I drew only 5 layers. I then realized, after continuous trials, that I can obtain all number from 0 to 32 in the tree diagram, and thus there are no numbers that are impossible to achieve.

Q3: Starting with a different number of pennies in the left drawer.

A3: First, it is obvious that the number has to be even because the pennies cannot be split if there are an odd number of pennies. I tried drawing sketches of tree diagrams for some even numbers I chose arbitrary, such as 48, 28, 36, 54, 100, and 20. At the same time, I also factorized the numbers into their prime factors, suspecting it would give some related clues. With that many examples written in front of me, I was soon able to see the pattern and hence the solution. One has to start with an even number of pennies, expressing the number as (odd*(2^n)), in which ‘n’ is a positive integer and ‘odd’ is any positive odd integer. The number of pennies one can obtain in a drawer is the multiples of ‘odd’ up until (odd*(2^n)).

Below is a picture of my rough work on the exercise paper. It might be difficult to understand since it is all over the place and there are no explanations or comments included. That, I think, is an area I need to improve upon. I need to be able to present my solution and procedures neatly, and maybe organize my procedures as I solve the problem.

More Proofs

  With two midterms coming up next week, this week’s slog is going to be short.

  This week in class we continued on with proofs. The examples we are tackling are getting consistently more difficult. For example, in Wednesday’s class, we focused on the question with “floor of x”. What caught my attention was when proving the result “floor of x > x - 1”, in one of the steps, we had to use “the contrapositive of the third point of the definition of floor of x”……After reviewing the material, I finally understood what the professor did there. In conclusion, I think I need more practice to be able to “see” what to do in a proof, because these methods, even though understandable, do not come intuitively.

    By the way we also proved the epsilon – delta definition of limits in class, which was a nice review for the upcoming MAT137 term test.

Term Test and Proofs

  This Wednesday we took our first term test. Honestly speaking the term test was not as difficult as expected and surprisingly similar to the practice test

  Now to proofs.

  Doing proofs is like solving mazes drawn on paper. You know where to start and where you want to end, all that is left is the “middle part”. You can work towards the result from the start (implication), or you can work backwards from the end (result). Doing proofs require a much better understanding of the problem and solution, and it is this that differs it from solving computational problems. One proof that we went through in class that I find interesting is Euclid’s proof that there are infinitely many prime numbers. The proof showed creativity with the use of contradiction and demonstrated full understanding of the subject by using the prime property that every integer >2 has a prime divisor. The proof was difficult to understand when reading it the first time, and I can only imagine how difficult it is to construct it from scratch.

Cheat Sheets

    The first term test is coming soon, and I am not sure if I am prepared or not. Luckily, we are allowed a cheat sheet for the test. Naturally, I tried to cram as much as I could on the two sided letter paper. However, I discovered that the act of copying notes on a cheat sheet actually helps the process of studying a lot, since I am summarizing all the notes. It also highlights my weaknesses, such as drawing Venn diagrams for the distributive laws. In fact, the process of making the cheat sheet might actually be more helpful than the cheat sheet itself during the exam. This is why I like math and logical reasoning. Unlike many other subjects, mathematics and logical thinking is a skill that will not be easily forgotten. More importantly, logic cannot be learned by memorization, which is why the cheat sheets are more reliable as a tool for studying than a lifesaver during the upcoming exam.