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Explore patterns in Pascal's Triangle!

In 1653, a french mathematician named Blaise Pascal described a triangular
arrangement of numbers corresponding to the probabilities involved in flipping
coins, or the number of ways to choose *n* objects from a group of
*m* indistinguishable objects.
The first seven rows of Pascal's Triangle look like:

1 n=0
1 1 n=1
1 2 1 n=2
1 3 3 1 n=3
1 4 6 4 1 n=4
1 5 10 10 5 1 n=5
1 6 15 20 15 6 1 n=6

Note that every number in the interior of the triangle is the sum of the
two numbers directly above it.
It turns out that Pascal's triangle holds many interesting numeric patterns.
One way of seeing some of these patterns is to pick a number *x* and
color all numbers in the triangle that are evenly divisible by *x*
with one color, and all the other numbers in the triangle with a second
color. To see as much of the pattern as possible, you need to be able to
see as many rows of the triangle as possible, but coloring a large number
of rows like this by hand is very boring and time consuming. A computer
can color 128 rows of the triangle in only a few seconds, so we can use
it to look at the results using many different divisors. The applet below
lets you choose the number that you want to use as a divisor. Then it colors
the first 128 rows of Pascal's triangle, coloring a square black if the
number that square represents is evenly divisible by the divisor you have
selected and red if it is not. To change the divisor, type in a new number
and click the "Set Divisor" button.

If you have a large monitor, here is a
version of the applet which displays 256 rows of the triangle.

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Here's something to investigate:

Look at the triangle using the divisors 3, 5, and 7.
Do you see a pattern? What do 3, 5, and 7 have in common?

Now try 9. Does the pattern continue?

Can **you** figure out what it is about 3, 5, and 7 that causes this
pattern, and why it doesn't continue at 9? Does it continue with other
numbers larger than 9?

Credits and related links

Copyright (c) 1997 Jeremy Baer

jbaer@countingstick.com