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One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.

Each number is just the two numbers above it added together (except for the edges, which are all "1").


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Diagonals

The first diagonal is, of course, just "1"s, and the next diagonal has the Counting Numbers (1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

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Horizontal Sums

What do you notice about the horizontal sums?

Is there a pattern? Isn't it amazing! It doubles each time (powers of 2).

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Fibonacci Sequence

Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence.

(The Fibonacci Sequence starts "1, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

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Heads and Tails

Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you "the odds" (or probability) of any combination.

For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "1,3,3,1" in Pascal's Triangle.

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The First 15 Lines

For reference, I have included row 0 to 14 of Pascal's Triangle


                                           1
                                        1     1
                                     1     2     1
                                  1     3     3     1
                               1     4     6     4     1
                            1     5     10    10    5     1
                         1     6     15    20    15    6     1
                      1     7     21    35    35    21    7     1
                   1     8     28    56    70    56    28    8     1
                1     9     36    84    126   126   84    36    9     1
             1     10    45    120   210   252   210   120   45    10    1
          1     11    55    165   330   462   462   330   165   55    11    1
       1     12    66    220   495   792   924   792   495   220   66    12    1
    1     13    78    286   715   1287  1716  1716  1287  715   286   78    13    1
 1    14     91   364   1001  2002  3003  3432  3003  2002  1001   364   91    14    1

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dinhi gid gagikang unsaom pagbalansi hang mga nomiro na ara gid sang taas, ginag aplay nila daw pera mid gang.




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