Pascal's Triangle is a mathematical pattern where each number is the sum of the two numbers directly above it. It was developed by ancient Chinese but Blaise Pascal discovered its patterns. The triangle exhibits several properties - the sum of each row is 2 to the power of the row number, certain numbers form squares, diagonals have special sum properties, and converting row numbers to single digits and raising 11 to that power reproduces the row. It contains many mathematical relationships worth exploring.

1. Pascal’s Triangle
By: Brittany Thomas

2. History &
Structure
Pascal's Triangle was originally
developed by the ancient
Chinese, but Blaise Pascal was
the first person to discover the
importance of all of the patterns it
contained. The tip of the triangle
is always a one, as well as all the
numbers on the outermost
diagonals. To get the other
numbers, you take any two
consecutive numbers and add
them. They create the number in
between them in the next row
down. The tip of the triangle is
considered the 0th element, the
second row is the 1st element,
and so on.

3. Sierpinski’s Triangle
The pattern Sierpinski’s
Triangle is formed when
you clearly distinguish the
odd numbers from the
evens. For example, in this
picture, the odd numbers
were colored black and the
even numbers remained
white. The pattern creates
more and more smaller
triangles as you color more
numbers.

4. The sum of the rows
The sum of the rows aren’t just
any random numbers. They also
form a pattern. The sum of a
row is equal to 2n when “n”
equals the number of the row.
As you can see from the
diagram, 2 to the 0th power
equals 1. The 0 represents that
it was the 0th row and in that
row there is only a one; 20
equals one .
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

5. Square numbers
A certain type of numbers in this
triangle are square numbers.
Square numbers are located in
the third diagonal. A Square
number is the sum of any two
consecutive numbers in the third
row of the triangle. Notice when
all the circled numbers in the
picture are added together, they
make a square number. For
example, 1+3=4 and 6+10=16. 4
and 16 are both perfect squares.

6. Hockey Stick Pattern
If a diagonal of numbers of any
length is selected starting at any of
the 1's bordering the sides of the
triangle and ending on any number
inside the triangle on that diagonal,
the sum of the numbers inside the
selection is equal to the number
below the last number of the
selection that is not on the same
diagonal itself.
1+6+21+56 = 84
1+7+28+84+210+462+924 = 1716
1+12 = 13

7. Magic 11’s
Another pattern discovered within the triangle is the Magic 11’s.
The idea behind this pattern is that you need to take each row and
convert it into a single number. The number is equal to 11 to the
nth power or 11n when n is the number of the row the multi-digit
number was taken from. For example, the third row (113) consists
of a 1,3,3, and a 1; 113 equals 1331.
Formula = Multi-Digit number Actual Row
110
= 1 1
Row 1 111
= 11 1 1
Row 2 112
= 121 1 2 1
Row 3 113
= 1331 1 3 3 1
Row 4 114
= 14641 1 4 6 4 1
Row 5 115
= 161051 1 5 10 10 5 1
Row 6 116
= 1771561 1 6 15 20 15 6 1
Row 7 117
= 19487171 1 7 21 35 35 21 7 1
Row 8 118
= 214358881 1 8 28 56 70 56 28 8 1