Problem Solving: Penny Piles

Unlike my recent posts, the title of this post will not start with "Week in Review." (I couldn't think of any better titles for my weekly posts.) I'll be writing about a problem solving exercise that our class did yesterday, called "Penny Piles."

The four steps to problem solving by Polya are:

1. Understand the Problem
2. Devise a Plan
3. Carry Out the Plan
4. Look Back

Using this approach, I'll explain how to solve this problem.

1. There are two drawers. One of the drawers is filled with 64 pennies, while the other one is empty. The question here is: Can things be arranged so that one of the two drawers has 48 pennies? This is where two operations are introduced: l and r.
• l: If the left drawer has an even number of pennies, half of that amount can be transferred to the right drawer. This operation will not work if the number of pennies on the left drawer is odd.
• r: If the right drawer has an even number of pennies, half of them can be transferred to the left. Once again, this will not work for odd amounts.
2. The plan is to consider the range [0, 64]. The goal is to arrange the pennies so that one of the two drawers has 48 pennies as well as another number within the range using the operations l and r from the previous step. To achieve this goal, a number of different possibilities can be mapped out in various ways.
3. At the start, the left drawer has 64 and the right drawer has 0. To execute the plan, half of the pennies in the left (32) can be transferred to the right using operator l. Now it's even. Both drawers have 32 pennies each. But one of the drawers must contain 48 pennies as per our problem. Operators l or r can be used in this case because 32 is an even number and the pennies are divided evenly. For example, if we transfer half of the pennies in the left drawer to the right, the result is 16 pennies in the left and 48 in the right. Therefore, it is possible for one of the drawers to have 48 pennies using arrangements, but the sequences are endless. Here is an example sequence using the l operator: 
• Start (left, right): 64, 0
• 32, 32
• 16, 48
• 8, 56
• 4, 60
• 2, 62
• Finish: 1, 63
4. To check the result, a tree diagram can be drawn, listing all the possibilities. Another way (similar to my approach) is to create an output sequence using only operators l, r, or a combination of both. Using an equation to derive the solution may also come in handy; for example, x + y = 64, where x and y represent pennies in the left and right drawers adding up to 64, respectively. To conclude, any number in the range [0, 64] can be achieved.