Elementary Arithmetic for the New Year? A Partition Problem for Middle Schoolers or for SATs

The following question(s) are appropriate for grades 5-12. How is this possible? Well, the mathematics needed to solve it is elementary! Of course, as with most math problems of this sort, the meaning/interpretation of the question is the challenge for most. Then there is the issue of going beyond the question to look for deeper mathematical meaning...

To help the younger student get started, we will begin with an example and then proceed with the question. Older students might need the same since the language of the question may be vague or difficult to comprehend:

The number 6 can be written as a sum of one or more odd positive integers in exactly 4 ways:
5+1, 3+3, 3+1+1+1, and 1+1+1+1+1+1. As you can see, we are not considering the order of the summands.

Also, 6 can be written as a sum of one or more different positive integers in exactly 4 ways:
6, 5+1, 4+2, 3+2+1. Again, different orders are not included in our list.

Anything of interest yet? Should students naturally raise their hands and ask questions about these two problems or do we have to cue them? At this point, the educator has many options. Here are a couple:

Pair of Problems:
List all ways to write 9 as a sum of one or more odd positive integers. How many ways?
List all ways to write 9 as a sum of one or more different positive integers. How many ways?
What do you notice?
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Investigation (in groups):
Make a table for positive integers from 1 through 10 as follows (I'm abbreviating the column headings). Also, include extra columns for the number of ways for each.

N..........N as a Sum of Odds...............N as a Sum of Different
1.........................1.......................................................1
2.......................1+1....................................................2
3.......
.
.
.
.
10

Comments:
(1) Can you think of at least 3 benefits from having students doing these?
(2) If class time does not permit further investigation, is it worth assigning these for homework, enrichment, extra credit, etc?
(3) Anyone imagine there might be some highly sophisticated ideas from number theory behind these simple questions? Anyone know who posed these kinds of questions originally and solved the general question?