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Answer:

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.

Each number is the numbers directly above it added together.

(Here I have highlighted that 1+3 = 4)

Patterns Within the Triangle

pascals triangle 1s, counting, triangular

Diagonals

The first diagonal is, of course, just "1"s

The next diagonal has the Counting Numbers (1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

Pascal's Triangle Symmetry

Symmetrical

The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.

pascals triangle powers 2

Horizontal Sums

What do you notice about the horizontal sums?

Is there a pattern?

They double each time (powers of 2).

pascals triangle powers 11

Exponents of 11

Each line is also the powers (exponents) of 11:

110=1 (the first line is just a "1")

111=11 (the second line is "1" and "1")

112=121 (the third line is "1", "2", "1")

etc!

But what happens with 115 ? Simple! The digits just overlap, like this:

pascals triangle powers 11b

The same thing happens with 116 etc.

pasca