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Exploring Pascal's Triangle
Patterns in Color
is an arithmetical triangle made up of staggered rows of numbers. Many patterns in the number
array can be seen more clearly using colors.
Applet by Multimedia Java Applications (MJA)
- Click on the "num on/off" button to show the numbers of Pascal's Triangle.
- To select a color, click on one of the 10 buttons below the triangle.
After you have done this, to color a hexagon click once inside it.
- Color each cell that contains an odd number.
- Click on the "num on/off" button to remove the numbers.
- The numbers in the cells of Pascal's triangle
are generated by adding two cells. With the numbers turned off you can see the pattern in the red and white cells.
- Notice that whenever two red cells are adjacent, the cell below is white. Can you explain why? Similarly, when two white cells are adjacent, the cell below is white. Can you explain why?
- Also notice that when one red cell and one white cell are adjacent, the cell below will be red. Do you think that this pattern will continue as you color more rows?
- The hexagon side length and number of rows can be adjusted using the "Side" and "Rows"
buttons. To see more than 11 rows, select 8 from the Side (pixels) menu above the triangle. Continue to color rows
to see whether the pattern will continue.
What can be learned?
- The operation of addition is commutative. Notice the symmetry in each row.
- The sum of two odd numbers will always be even. Two adjacent
red cells have a white cell below.
- The sum of two even numbers will always be even. Two adjacent
white cells have a white cell below.
- The sum of an odd number and an even number will always be odd. A red cell adjacent
to a white cell will have a red cell below.
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