Time for MATH MINUTE! (provide your favorite theme music here). 

         Get out your paper & pencil, because I have a new puzzle for you!!

Responses/Hints: PUZZLE #90 Polygonal numbers

       These results can be found in The Book of Numbers, by J. Conway and R. Guy.

 

A.    Given an infinite list of numbers beginning with 1, choose the 1st number 1, then compute the sum of the 1st two numbers, then the sum of the 1st three numbers, and so on, as follows:

 

       The list: 1, 1, 1, 1, . . .  gives the counting line numbers: 1, 2, 3, 4, . . .

       The list: 1, 2, 3, 4, 5, . . . gives the triangular numbers: 1, 3, 6, 10, . . .

 

       What list of numbers beginning with 1 would give the square numbers: 1, 4, 9, 16, . . . ?

       The objective is to generate the S (square) numbers 1, 4, 9, . . . , so it is obvious (?) that the 1st few numbers in the list must be 1, 3, & 5.

       This suggests the following pattern (compare above for L (line) and T (triangular) numbers), try the (odd) numbers where 2 is added each time to get the next number: 1, 3, 5, 7, . . .   .

 

       Are any of the square numbers also triangular numbers?

       You have to write out a few of each type of numbers to see this:

              T numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, . . .

              S numbers: 1, 4, 9, 16, 25, 36, 49, . . .

       So we see that 36 is both a T number and a S number.

       I wonder if there are any others?

      

       Can this method be extended to larger polygonal numbers? How?

       Yes, the next set of generators is the list of numbers that you get by adding 3 each time: 1, 4, 7, 10, 13, . . .  .

       And the numbers that they generate are the pentagonal numbers:

              P numbers: 1, 5, 12, 22, 35, . . .  .

       Notice that each P number is 1/3 of a T number? Is this true in general?

       You can repeat this method for higher polygonal numbers.

 

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?

       A 5x5 triangle:

                            

       In fact you could get the triangular numbers directly from the picture without first computing them? Explain how this would be done.

       Take a line of red dots in the L numbers and add a green triangle which is 1 less in size, like we did above for the red L number 5 and the green 4x4 triangle.

 

       For the square numbers what would the dot picture be, if you just used the dot picture for the triangular numbers as a starting point?

       You need to add a small (blue) triangle:

      

       Can this process be extended to the higher polygonal numbers?

       How?

       Yes, just repeat the above patterns.

       For pentagonal numbers you have to add a certain number of yellow dots:  

      

       Notice how the previous pentagon can be worked into the next one.

 

       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/MM1.1.htm 

to see the results every Friday.

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