Triangle Numbers in Pascal’s Triangle

One of the most famous patterns that can be found in Pascal's triangle is the triangle numbers, which can be found in the diagonals of Pascal's Triangle. In this article, I explain in a variety of different ways what triangle numbers are, as well as showing exactly where triangle numbers can be found in Pascal's triangle.

Triangle numbers can be found by looking at the second diagonal in Pascal's triangle when it is drawn centrally. However, by left justifying the numbers in Pascal's triangle, as in the diagram below, they can be found in the third column:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

The sequence in the third column begins 1,3,6,10,15,21. This is the sequence we’re interested in. To help explain what this sequence is, I have shown two different ways of thinking of it below:

Method 1: The nth triangle number is the sum of all the positive whole numbers up to n. This may sound confusing, but I will give a couple of examples. 21 is the seventh triangle number, because it is the sum of 1+2+3+4+5+6+7. So what about the tenth number in the sequence? It is 1+2+3+4+5+6+7+8+9+10, which if you work it out is 55. Incidentally, you may have been given the task at school before of adding up all of the numbers from 1 to 100 or 1 to 1000. What you are really being asked to do is find the 100th or 1000th triangle number (this can be done quickly in a variety of sneaky ways - unfortunately however, calculating all the rows of Pascal's triangle and looking down the third column is NOT one of these methods!)

Method 2: This method is much more exciting. It's a hands on, visual way of understanding triangle numbers, and it also explains where this sequence gets its name from. You will need some coins or counters (or a pen and paper). What you need to do is arrange your counters in triangles. You want to create an equilateral triangle (one with the same number of counters on each side), and you need to "fill in" your triangle with counters, not just put counters round the edges of the triangle. Now, if you were trying to find the 6th triangle number, you would need to create a triangle with 6 counters on each side. Once you have done that, you just have to count the number of counters you have used. You should find you get 21, which is correct (this can be checked with method 1 or Pascal's triangle). You can try this out with any sized triangle of counters and it should always work.

Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;}