Time for MATH MINUTE!
(provide your favorite theme music here). Get out
your paper & pencil, because I have a new puzzle for you!!
PUZZLE #79 How many possible sums?
Consider
the number 4.
It
can be written as a sum of positive numbers in 8 ways:
1+1+1+1,
1+1+2, 1+2+1, 2+1+1, 1+3, 3+1, 2+2, 4.
We
have just determined the compositions of 4 (also called the ordered
partitions of 4). The individual sums are called composition numbers.
Compositions
have many uses in not only mathematics, but also in computer science, and in the
sciences.
A.
Can you find the compositions of 5?
Did
you get 16?
Find
the compositions of 1 (this is trivial, it is 1), 2, and 3.
Do
you see a pattern here in the total number of the compositions for each of the
numbers 1, 2, 3, 4, 5?
Can
you predict how many composition numbers are in the compositions of 6?
Can
you find them? There are quite a few.
B. If we ignore the composition numbers which use the same numbers (like 1+1+2,
1+2+1, and 2+1+1 in the compositions of 4) we get the partitions of the
number (for 4 we have: 1+1+1+1, 1+1+2, 1+3, 2+2, 4).
So
there are 5 partition numbers for 4.
Find
the partitions for 5, 6.
Finding
a pattern for the total number of partitions of a given whole number is much
harder (check Wikipedia).
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
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).