Triangular Numbers

Date: 11/15/95 at 9:59:19
From: East Lansing
Subject: triangular numbers

Dear Dr. Math,

We are 2nd and 3rd graders and have been looking at the patterns 
triangular numbers make.  We have been having fun but we don't know 
why 1 is a triangular number.  It doesn't look like a triangle to us.  Our 
teacher doesn't really know either.  Our math consultant, Mrs. Yost, said 
we could ask you. 

Ms. Graven's class

Date: 11/15/95 at 15:11:2
From: Doctor Ken
Subject: Re: triangular numbers

Hello!

The triangular numbers are pretty cool, aren't they!  Perhaps I can tell
you a little bit about why we consider 1 a triangular number by talking 
about _square_ numbers a little bit.

Here are some pictures of the square numbers:

*    * *    * * *     * * * *      * * * * * 
1    * *    * * *     * * * *      * * * * * 
      4     * * *     * * * *      * * * * * 
              9       * * * *      * * * * * 
                        16         * * * * * 
                                      25

and so on.  Notice that these numbers are the same as the "perfect 
squares," those numbers that can be written as some number times itself.  
For instance, 9 = 3x3, and 16 = 4x4.  Well, in the same way, 1 = 1x1.  
So we like to call 1 a square number.  It's really just kind of a matter 
of taste.

Now look at the pictures of the squares.  We've got a square with 3 
stars on each side, a square with 2 stars on each side, heck, why can't 
we have a square with only 1 square on each side?  That's the square 
with only 1 total star.

In a similar way, we can look at the triangular numbers:

  *      *        *          *             * 
  1     * *      * *        * *           * * 
         3      * * *      * * *         * * * 
                  6       * * * *       * * * * 
                            10         * * * * * 
                                          15

You can think of that first one as a triangle with only 1 star on each 
side.  It just so happens that that makes the whole triangle have only 
one star total.  

So the weird part is that a single star sitting there on a piece of 
paper can be a square, a triangle, a pentagon, a hexagon, whatever the 
heck you want it to be!  It's like magic.  The magic single star!

-Doctor Ken,  The Geometry Forum