Another relation in Pascal's Triangle is the sum of the terms in each row of the triangle.
Simply put: the sum of the numbers in the nth row minus 1 == the total number of moves required to move an ordered stack of nth discs from one column to another.
(Note: The starting row in Pascal's Triangle is the 0th row)
Example:
If we had 0 discs, we need 0 moves. ((1) - 1)
If we had 1 disc, we need 1 move. ((1+1) - 1)
If we had 2 discs, we need 3 moves. ((1+2+1) - 1)
If we had 3 discs, we need 7 moves. ((1+3+3+1) - 1)
jstaahl
commented
10 years ago
There are some interesting relationships between Pascal's Triangle, Sierpinski's Triangle, and the Tower of Hanoi.
For Pascal's Triangle, if every number were in a square and we were to shade in that square if the number were odd, we would observe a pattern that is the Sierpinski's Triangle. See link: https://www.zeuscat.com/andrew/chaos/pascal.sierpinski.clear.gif
Another relation in Pascal's Triangle is the sum of the terms in each row of the triangle. Simply put: the sum of the numbers in the nth row minus 1 == the total number of moves required to move an ordered stack of nth discs from one column to another. (Note: The starting row in Pascal's Triangle is the 0th row) Example: If we had 0 discs, we need 0 moves. ((1) - 1) If we had 1 disc, we need 1 move. ((1+1) - 1) If we had 2 discs, we need 3 moves. ((1+2+1) - 1) If we had 3 discs, we need 7 moves. ((1+3+3+1) - 1)
Create a Swing GUI to observe this phenomenon.
~estimated 250