Exercises on Numerical Partitions

Part 1

1. List all the ways to make $0.50 in change using pennies, nickels, dimes and quarters. Without listing the coin changes, do you think there are fewer, more, or the same number of ways to make $0.51 in change? Explain your answer.

2. Based on your explanation in #1, what is the next larger amount for which the number of ways to make change will be different from the number for $0.50? Explain your answer.

3. Repeat the lists of numerical partitions of the numbers 1 through 7 and add the lists of numerical partitions of 8, 9, and 10.

4. Using your lists from #3, find the number of partitions of n that use only even numbers and the number of partitions that use only odd numbers. Make a table to summarize the data. What observations can you make about the data?

5. Using your lists from #3, find the number of partitions of n that do not repeat any digits. Make a table to summarize this data. What observations can you make?

6. Construct Ferrer’s diagrams of the 15 partitions of 7. Identify the partitions that form conjugate pairs and any partitions that are self-conjugate.