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#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 nth 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 = 9² 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 = n²+n .