How Triangular Numbers make Square Numbers

 Here are the first 4 triangular numbers.  

*   1

                  * 
                  *   *   3  

*
*   *      
*   *   *  6

                             *
                             *   *
                             *   *   *
                             *   *   *   *  10

If you add successive pairs of triangular numbers together (e.g. 1 + 3 = 4) , you will get a square number

3 + 6 = 9, a square number. To see this visually “rotate” the 3 so that it fits in the upper right hand corner of the 6. You will then have a 3 x 3 square, or 9. 

You can make more triangular numbers by adding another column to the left of the last number you made. So if you add a column 5 high to the left of the 4 column of the triangular number 10 you’ll get 15 the next triangular number.

You can keep doing this indefinitely.