Pascal’s TrianglePascal’s Triangle

By: Brittany Thomas

History & History & StructureStructure

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

Sierpinski’s TriangleSierpinski’s Triangle

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

The sum of the rowsThe sum of the rows

The sum of the rows aren’t just The sum of the rows aren’t just any random numbers. They also any random numbers. They also form a pattern. The sum of a form a pattern. The sum of a

row is equal to 2row is equal to 2nn when “n” when “n” equals the number of the row.equals the number of the row.

As you can see from the As you can see from the diagram, 2 to the 0diagram, 2 to the 0thth power power equals 1. The 0 represents that equals 1. The 0 represents that it was the 0it was the 0thth row and in that row and in that row there is only a one; 2row there is only a one; 20 0

equals oneequals one ..

2200 = 1 = 12211 = 1+1 = 2 = 1+1 = 22222 = 1+2+1 = 4 = 1+2+1 = 42233 = 1+3+3+1 = 8 = 1+3+3+1 = 82244 = 1+4+6+4+1 = 16 = 1+4+6+4+1 = 16

Square numbersSquare numbers

A certain type of numbers in A certain type of numbers in this triangle are square numbers. this triangle are square numbers. Square numbers are located in Square numbers are located in the third diagonal. A Square the third diagonal. A Square number is the sum of any two number is the sum of any two consecutive numbers in the third consecutive numbers in the third row of the triangle. Notice when row of the triangle. Notice when all the circled numbers in the all the circled numbers in the picture are added together, they picture are added together, they make a square number. For make a square number. For example, 1+3=example, 1+3=4 4 and 6+10=and 6+10=1616. 4 . 4 and 16 are both perfect squares.and 16 are both perfect squares.

Hockey Stick PatternHockey Stick Pattern

If a diagonal of numbers of any If a diagonal of numbers of any length is selected starting at any of length is selected starting at any of the 1's bordering the sides of the the 1's bordering the sides of the triangle and ending on any number triangle and ending on any number inside the triangle on that diagonal, inside the triangle on that diagonal, the sum of the numbers inside the the sum of the numbers inside the selection is equal to the number selection is equal to the number below the last number of the below the last number of the selection that is not on the same selection that is not on the same diagonal itself.diagonal itself.

1+6+21+56 = 841+6+21+56 = 841+7+28+84+210+462+924 = 17161+7+28+84+210+462+924 = 17161+12 = 131+12 = 13

Magic 11’sMagic 11’s

Another pattern discovered within the triangle is the 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 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 convert it into a single number. The number is equal to 11 to the

nnthth power or 11 power or 11nn when n is the number of the row the multi-digit when n is the number of the row the multi-digit number was taken from. For example, the third row (11number was taken from. For example, the third row (1133) consists ) consists of a 1,3,3, and a 1; 11of a 1,3,3, and a 1; 113 3 equals 1331.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 1Row 7 11

7= 19487171 1 7 21 35 35 21 7 1

Row 8 118

= 214358881 1 8 28 56 70 56 28 8 1