When you first look at the image above, your first reaction is "oh those are just some dumb ol' number strung together in a random order." When you look closer however, you will notice all the cool patterns and tricks of this famous little triangle called Pascal's Triangle.
The first pattern that you will notice is that every new row is made up of sums from the previous row. For example, row 4 is 14,641 because you always start a new row with 1, then 1+3 (from row three) = 4, 3+3 (from row three) = 6, 1+3 (from row three) = 4. Finally, you always end a row with 1.
The second pattern that I noticed was that the blue line (side image) increases by one all the way down the diagonal to 13. And similarly, the green line (side image) increase by adding numbers in numerical order. For example, 1+2 = 3, 3+3 = 6, 6+4 = 10 and so on and so on down the diagonal to 78.
The third pattern is a little harder to discover and is kinda hidden within the triangle. If you take the sum of each row, you will see it will be double the previous rows. I'll show you what I mean:
ROW 0: 1
ROW 1: 1+1=2
ROW 2: 1+2+1=4
ROW 3: 1+3+3+1=8
ROW 4: 1+4+6+4+=16
...
ROW 8: 1+8+28+56+70+56+28+8+1=256
ROW 9: 1+9+36+84+126+126+84+36+9+1=512
...
ROW 12: 1+12+66+220+495+792+924+792+924+792+495+220+66+12+1=4,096
ROW 13: 1+13+78+286+715+1287+1716+1716+1287+715+286+78+13+1=8,192
Of course there are more patterns and tricks in the triangle, but if I mentioned them all then this would be a very long blog. I picked the patterns I thought were interesting and cool to share. But there are more out there so what can you find?
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