French mathematician, Blaise Pascal (1623-1662) created a system of numbers containing many mathematical patterns and sequences. With mathematical patterns, we can learn problem solving strategies. This triangle is known today as Pascal's Triangle. Throught the world, people have used this triangle to solve mathematical problems.
Some patterns that can be found in Pascal's triangle include:
The outer two diagonals of the triangle. From this we can see that the outer diagonal only conatins the number 1, the inner diagonal from there, is an arithmatic sequence from 1 to 13.
Next, we have the Hockey Stick Pattern. 
This is called the hockey stick pattern, because if you look the numbers circled, it forms the shape of a hockey stick. With this pattern, we can see that If a diagonal of numbers are selected starting with any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that same diagonal, the sum of the numbers inside the selection is equal to the number below the last number in the selection that is not on the same diagonal itself.
The sums of the rows add up to the power of 2, as seen in this diagram. If we look at the first row, 1, 1+0=1; 2^0. The next row: 1+1=2; 2^1
1 2 1: 1+2+1=4; 2^2
1 3 3 1: 1+3+3+1=8; 2^3
1 4 6 4 1: 1+4+6+4+1=16; 2^4
1 5 10 10 5 1: 1+5+10+5+1=32; 2^5
and so on, as illustrated in this figure.
I think the coolest pattern of Pascal's Triangle is the "hockey stick pattern". I never would have found that one. I probably wouldn't have found "the sum of the rows" pattern either. I find the whole concept of the triangle kind of fascinating.
ReplyDeleteI find the Pascal's Triangle to be very interesting. It is so interesting because there are so many cool patterns. Who would have thought there would be a Triangle that is interesting???
ReplyDeleteOrganic patterns are beautiful. Your blog inspires me to wonder what I could discover, if i just looked hard enough at the natrual patterns around us all...
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