A great program! Make an unlimited amount of money as
long as you're connected to the net.
Be sure to enter "wintonp" as your referrer :-)
And Its Patterns
Pascal's Triangle was originally developed by the ancient Chinese, but Blaise Pascal was
the first person to discover all of the patterns it contained. On this page, I explain
how the Triangle is formed, and more importantly, many of its patterns.
At the tip
of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two
1's, both formed by adding the two numbers above them to the left and the right, in this case
1 and 0 (all numbers outside the Triangle are 0's).
Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1.
And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the
rows of the triangle go on infinitly. A number in the triangle can also be found by nCr (n Choose r)
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 element number 1, the next three is the 2nd element, and
the last 1 is the 3rd element. The formula for nCr is:
! 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.
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
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.
Fibonnacci's 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
Try the formula out below (requires a VBScript capable browser)
Triangular Numbers are just one type of polygonal numbers. See the section on
Polygonal Numbers for an explaination of polygonal and
triangular 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 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 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. (Look at
the table below for a clearer explaination). My friends' formula (the Shi-Cheng formula)
for the nth x-gonal number (for example: the 2nd 3-gonal,
or triangular number) is:
If x is even, then:
y = x/2 - 1 and the formula is n+y(n2-n)
If x is odd, then:
y = (x-1)/2 and the formula is (-(n2)+3n+2n2y-2ny)/2
These formulas work fine, but I think my own formula (the Winton formula) is much less convoluted,
and is based on the fact that to find the nth x-gonal number, you multiply
the number in the 3rd diagonal in the nth row by x-2, and
then add the number in that same row but in the 2nd diagonal. Therefore:
((n2-n)/2) × (x-2) + n
Find the nth x-gonal number using the Winton formula:
Find the nth x-gonal number using the Shi-Cheng formula:
As you may have noticed, the numbers in the chart above are actually the tip of the
right-angled form of Pascal's Triangle, except the preceeding 1's in each row are missing.
The circular figures are formed by simply placing a number of points on a circle and then
drawing all the possible lines between them. This chart shows that for a figure with
n points, all you need to do is look at the nth row of the Triangle
in order to find the number of points, line segments, and polygons in
the figure with ALL of their vertices on the circle.
When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle
are filled in (black) and the rest (the evens) are left blank (white), the recursive
Sierpinski Triangle fractal is revealed (see figure at near right), showing yet another
pattern in Pascal's Triangle. Other interesting patterns are formed if the elements not divisible
by other numbers are filled, especially those indivisible by prime numbers.
Go here (it works now) to download programs that calculate
Pascal's Triangle and then use it to create patterns, such as the detailed, right-angle
Sierpinski Triangle at the far right.