Time for MATH MINUTE! (provide your favorite theme music
here).
Get out
your paper & pencil, because I have a new puzzle for you!!
PUZZLE #90 Polygonal numbers
These results can be found in The
Book of Numbers, by J. Conway and R. Guy.
-
I have rewritten this to make it clearer (I hope).
A. Given an
infinite list of numbers beginning with 1, choose the 1st number 1,
then the sum of the 1st two numbers, then the sum of the 1st
three numbers, and so on to get a new list:
1,
1, 1, 1, . . . gives the counting line
numbers 1, 2, 3, 4, . . .
1,
2, 3, 4, 5, . . . gives the triangular numbers 1, 3, 6, 10, . . .
What
list of numbers would give the square numbers 1, 4, 9, 16, . . . ?
Are
any of the square numbers also triangular numbers?
Can
this method be extended to larger polygonal numbers? How?
B. We can
also picture these results geometrically by using dots as follows:
For
the counting line numbers 1, 2, 3, 4, . . . , we have:
1 2 3 4
For
the triangular numbers 1, 2, 6, 10, . . . , we have:
1 3 6 10
Notice
how the dot picture for the triangular numbers can be obtained from the dot
picture for the counting line numbers by adding a triangle?
What
size triangle will give the next dot picture for the triangular number 15?
In
fact you could get the triangular numbers directly from the picture without
first computing them? Explain how this would be done.
For
the square numbers what would the dot picture be, if you just used the dot
picture for the trianglular numbers as a starting point?
Can
this process be extended to the higher polygonal numbers? How?
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/MM1.1.htm
to see the results every Friday.
See you next time on MATH MINUTE! (theme music fades out here).