Question: In how many ways can change amounting to a given amount be made?

1. Using pennies, nickels, dimes and quarters, make a list of all the ways to make $0.12. Repeat for $0.27, $0.43.

Each of the possible ways of using coins to make a given amount of change is called a coin changing. These are special cases of what are known as numerical partitions of any positive integer. A numerical partition of an integer n is a sequence of integers called parts whose sum is n. For example, a partition of 14 is 6+4+2+1+1.

Note that the parts are listed from largest to smallest.

2. Find all the numerical partitions of the numbers 1 through 7.

The number of partitions of a given number n is denoted by p(n).

3. Make a table of your results in #2 with columns listing n and p(n).

n	p(n)
1
2
3

Every numerical partition of a number n corresponds to a unique "Ferrer's diagram". This is formed by an arrangement of n dots on a grid where each part in the partition is represented by placing the same number of dots in a row. Below is the Ferrer's diagram for the partition 7+4+4+1+1+1 of n = 18.

7
4
4
1
1
1

4. Draw the Ferrer's diagrams for the partitions of 5. Repeat for the partitions of 6.

Note: Two Ferrer's diagrams are conjugate pairs if one diagram can be transformed into the other by changing rows into columns and columns into rows. A Ferrer's diagram is self-conjugate if changing rows and columns leaves the diagram unchanged.

5.  Use your Ferrer’s diagrams to find all the conjugate pairs and self-conjugate partitions of 5. Repeat for 6.