#math trivia for #February14: 45 is the 9th triangular number: 45 = 1+2+3+…+9. How do you make a 9×9 square out of 45-unit triangles?

— Burt Kaliski Jr. (@modulomathy) February 15, 2012

The *n*th **triangular number** has the form 1+2+3+…+*n*, and is so named because it measures the “units” in a triangularly shaped figure with *n* rows with widths increasing from 1 to *n*. The 9th triangular number corresponds to this figure with 9 rows:

*

**

***

****

*****

******

*******

********

*********

Each “*” is one unit. All together, there are 45 units in this figure.

The number of units in the figure with *n* rows is:

1 + 2 + 3 + … + *n* = *n*(*n*+1) / 2 .

To see this, consider that the “average” row has (*n*+1)/2 units, and there are *n* rows.

Following a similar construction, the 9th **square number** measures the number of units in a square shaped figure with 9 rows, each with width 9, i.e., a 9×9 square:

*********

*********

*********

*********

*********

*********

*********

*********

*********

The figure has 81 = 9^{2} units, as one would expect.

How do you make a 9×9 square out of 45-unit triangles? Overlap them along the diagonal:

\********

*\*******

**\******

***\*****

****\****

*****\***

******\**

*******\*

********\

The overlap illustrates the relationship

2*45 = 81 + 9

or more generally

*n*(*n*+1)/2 + *n*(*n*+1)/2 = *n*^{2}+*n* .