Problem-Solving Episode

Dear Blog,

I am going to walk you through a problem-solving episode, using Polya's problem-solving technique. This problem is one we tried during a lecture. It is called Penny Piles. The problem states that you are sitting in front of two drawers. The left drawer contains 64 pennies and the right drawer contains 0 pennies. The problem is to figure out if you can arrange the drawers so that  one has 48 pennies using the following operations of l and r.

l: if the left drawer has an even number of pennies, you may transfer half to the right drawer
r: if the right drawer has an even number of pennies, you may transfer half to the left drawer.

After doing this, you must choose a number other than 48 in the range [0,64] and try to arrive with that many pennies in one drawer with the same initial position. It then asks if there are any numbers in that range that are impossible to achieve.

1. Understand the problem:

At first glance, this problem was super confusing, but after re-reading the problem a few times, I fully understood it. Basically you start with 64 pennies in one drawer and 0 in the other. You are allowed to take half of the pennies from the drawer with 64 and put them in the other, leaving you with 32 and 32. You can continue doing this with both drawers as long as they contain an even number of pennies.

2. Devise a plan:

I started out by trying to make one drawer have 48 and one have 16, satisfying the first condition.

Steps	Left	Right
0	64	0
1	32	32
2	48	16

I will then try to get other numbers in the range [0,64], by continuously dividing numbers by two and trying to arrive at new numbers.

3. Carrying out the Plan:

Simply through trial and error, I realize that every number in the range [0,64] is possible to achieve. (This took a while).

4. Looking Back

Since this step involves looking back, I did just that, and I realized, there has to be a much easier way to solve this problem without going through every possible number combination. So, I devised a new plan. I tried many times to think of an equation for this problem, but have so far failed. But this problem has stuck with me and every now and then I still try. I have drawn out many branches of combinations and it seems to me that it if this is a function, it is an exponential function, and it may be important to start at an even square, like 64 or 144. For the time being, I am stumped, but this is a very interesting problem.

Thank you,

-J.M