Problem Solving Exercise :

Penny Piles Exercise:

The PDF version of the problem can be found here: http://www.megaupload.com/?d=2ZM1BL0K

1) Understand the Problem

This dazzling question that involves pennies and drawers can be easily simplified as dividing 64 (or any even number) by half and produing a certain number by adding quotients.

2) Devise a Plan
First try to 48 as indicated in the question. If that's possible, try some other numbers to see if they are possible to produce. Record the numbers that have been sucessfull produced to see if there is any pattern.

3) Carry out the Plan
48 can be obtained by performing the following steps:

Furthurmore, by furthur developing the above diagram, we have:

As we can readily see from the above nicely-laid-out diagram, every number from 0 to 64 can be represented by moving half of the pennies from one drawer to another. The underlying reason for this is that any positive integer can be represented as additions of several 2^n, where n \in N. For example, 115 = 2^6 + 2^5 + 2^4 + 2^1 + 2^0.

4) Conclusion

Hence, as long as the number of pennies in the left drawer can be represented as 2^n, n \in N, we can always produce any number of pennies in the range of [0, 2^n] by swapping half of the pennies from one drawer to the other.