One of the beauties of mathematics is its ability to display relationships and patterns of numbers. The French mathematician, Blaise Pascal, was a particular proponent of this branch of mathematics and this is demonstrated no more so, than in Pascal's Triangle.

Pascal's Triangle is formed by drawing a triangular arrangement of numbers commencing with the number 1 at its apex, with each subsequent row commencing with the number 1 and then each adjacent number being the sum of the two numbers in the row immediately above. This is best illustrated in the following link:

https://en.wikipedia.org/wiki/Pascal%27s_triangle#/media/File:Pascal_triangle_small.png

The sequence continues ad infinitum but this very simple structure contains some very interesting properties.

The leading diagonal is not particularly exciting, just being a row of 1's and the second leading diagonal is the list of natural numbers. So far, so good.

The third leading diagonal is the list of triangular numbers. Think of these as the number of snooker balls required to make up various sized triangles. The fourth leading diagonal is the list of tetrahedral numbers. These are the number of snooker balls required to make up pyramids of various sizes with triangular bases, and so on.

Now what happens if we sum each row in Pascal's Triangle? The result is we obtain the powers of 2, i.e. 1, 2, 4, 8, 16, etc.

The rows of the triangle also show a relationship to the powers of 11. The first row is 11^{0}=1. The second row is 11^{1}=11. The third row is 11^{2}=121 and so on. The pattern may appear to break down when we get to double digits appearing in the triangle, as in the fifth row. However, all that is required is to carry over the tens digit to the number on its left. So in the fifth row, 11^{5}=161051.

The second diagonal also reveals a list of their perfect squares shown in the third diagonal. So, for example, the square of the number two in the second diagonal, is the sum of the first two numbers in the third diagonal. The square of the number 4 in the second diagonal, is the sum of the adjacent numbers 6 and 10 in the third diagonal.

There are many, many more patterns and relationships revealed by Pascal's triangle, such as those relating to binomial expansion, that are worth exploring but this is just one of the numerous properties of mathematics that makes it such a fascinating subject to explore.