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.