Time for MATH MINUTE! (provide your favorite theme music here). Get out your paper & pencil, because I have a new puzzle for you!!

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Responses/hints: PUZZLE #79 How many possible sums?

A. Compositions:
1: 1

2: 1+1, 2

3: 1+1+1, 1+2, 2+1, 3

4: 1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 1+3, 3+1, 2+2, 4

5: 1+1+1+1+1, 1+1+1+2, 1+1+2+1,1+ 2+1+1, 2+1+1+1, 1+2+2, 2+1+2, 2+2+1, 1+1+3, 1+3+1, 3+1+1, 2+3, 3+2, 1+4, 4+1, 5

6: 1+1+1+1+1+1, 1+1+1+1+2, 1+1+1+2+1,1+1+ 2+1+1, 1+2+1+1+1, 2+1+1+1+1, . . . and so on

Total number of compositions:

1: 1

2: 2

3: 4

4: 8

5: 16

6: 32

7: 64 (?)

Did you notice that each total number of compositions was a power of 2?

The formula for the total number of compositions of a positive number n is 2^n-1.

B. Partitions: (these are easier?)

1: 1

2: 1+1, 2

3: 1+1+1, 1+2, 3

4: 1+1+1+1, 1+1+2, 1+3, 2+2, 4

5: 1+1+1+1+1, 1+1+1+2, 1+2+2, 1+1+3, 2+3, 1+4, 5

6: 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 3+3, 1+2+3, 1+1+4, 2+4, 1+5, 6

So the total number of partitions for each number is:

1: 1

2: 2

3: 3

4: 5

5: 7

6: 11

Is there a pattern? The interesting thing here is that although calculating partitions is fairly easy, finding a formula for the total number is not so simple. I can only refer you to the website at Wikipedia:

http://en.wikipedia.org/wiki/Partition_(number_theory)_

Have fun!

Send your comments, ideas and solutions before Monday to the email below, and in the subject line be sure to put MM in the subject line

parksjm@potsdam.edu

Visit us here online at:

http://www2.potsdam.edu/parksjm/MM1.htm

to see the results every Friday.

         See you next time on MATH MINUTE! (theme music fades out here).