Pascals triangle How Pascals Triangle is built. On the yop of Pascal's Triangle is the number 1, that is the zeroth row. The first row has two 1's, both made by adding the two numbers above them to the left and the right, in this case 1 and 0 because all numbers outside the Triangle are 0's. Do the same to make the second row (1+1=2). And the third (1+2=3 and 2+1=3). Doing this, the rows of the triangle go on infinitly. A number in the triangle can also be found by nCr where n is the number of the row and r is the element in that row. For example, in row 3, 1 is the zeroth element, 3 is the first element, the next 3 is the second element, and the last 1 is the 3rd element. The formula for nCr is: n! / r!(n-r)! ! means factorial, or the preceeding number multiplied by all the positive integers that are smaller than the number. 5! = 5 × 4 × 3 × 2 × 1 = 120. The Sums of the Rows The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example: 20 = 1 21 = 1+1 = 2 22 = 1+2+1 = 4 23 = 1+3+3+1 = 8 24 = 1+4+6+4+1 = 16 Prime Numbers If the 1st element in a row is a prime number, all the numbers in that row, excluding the 1's can be divided by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7. 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 end of the selection that is not on the same diagonal itself. If you don't understand that, look at the drawing. 1+6+21+56 = 84 1+7+28+84+210+462+924 = 1716 1+12 = 13 Magic 11's If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), 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. Row # Formula = Multi-Digit number Actual Row Row 0 110 = 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 Fibonnaccis Sequence Fibonnaccis Sequence can also be located in Pascal's Triangle. The sum of the numbers in the consecutive rows shown in the diagram are the first numbers of the Fibonnacci Sequence. The Sequence can also be formed in a more direct way, very similar to the method used to form the Triangle, by adding two consecutive numbers in the sequence to produce the next number. The creates the sequence: 1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . . . . The Fibonnacci Sequence can be found in the Golden Rectangle, the lengths of the segments of a pentagram, and in nature, and it decribes a curve which can be found in string instruments, such as the curve of a grand piano. The formula for the nth number in the Fibonnacci Sequence is: Triangular Numbers Triangular Numbers are just one type of Polygonal Numbers. The triangular numbers can be found in the diagonal starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the fourth is 10, and so on. Square numbers Square Numbers are another type of Polygonal Numbers They are found in the same diagonal as the triangular numbers. A Square Number is the sum of the two numbers in any circled area in the diagram. (The colors are different only to distinguish between the separate "rubber bands"). The nth square number is equal to the nth triangular number plus the (n-1)th triangular number. (Remember, any number outside the triangle is 0). The interesting thing about these 4-sided polygonal numbers is that their name explains them perfectly. The very first square number is 02. The second is 12, the third is 22 (4), the fourth is 32 (9), and so on. Read on to the Polygonal Number section to learn more. Polygonal Numbers Polygonal Numbers are really just the number of vertexes in a figure formed by a certain polygon. The first number in any group of polygonal numbers is always 1, or a point. The second number is equal to the number of vertexes of the polygon. For example, the second pentagonal number is 5, since pentagons have 5 vertexes (and sides). The third polygonal number is made by extending two of the sides of the polygon from the second polygonal number, completing the larger polygon, and placing vertexes and other points where necessary. The third polygonal number is found by adding all the vertexes and points in the resulting figure. If this is hard to understand, try clicking here!