Pascal's Triangle










     Now knowing how to construct Pascalís Triangle, one may wish to look deeper into its rows and diagonals to spot some of its greatest mysteries.  One of the triangleís first mysteries to be spotted is that the numbers of the array are symmetric.  In other words, if one were to fold the triangle across its altitude, it would be clear to the observer that the numbers match.  As one peers deeper into the triangleís numbers, patterns such as the powers of two, the powers of eleven, the triangular numbers, and Fibonnaciís sequence may be noticed.  To allow for an appreciation of some of its mysteries, this discussion will focus on several of the patterns, such as these, found within Pascalís Triangle.  Yet, in order to appreciate this mathematical beauty fully, I encourage all readers to try to uncover more of its secrets and patterns for themselves.

    One of the patterns of Pascalís Triangle is displayed when one finds the sums of the rows.  In doing so, it can be established that the sum of the numbers in any row equals 2n, when n is the number of the row.  For example: 

                                          1                         =   1  =  20 
                                          1 + 1                   =   2  =  21 
                                          1 + 2 + 1             =   4  =  22 
                                          1 + 3 + 3 + 1       =   8  =  23 
                                          1 + 4 + 6 + 4 + 1 = 16  =  24.

    There is also an interesting pattern in the triangle dealing with prime numbers.  For any prime numbered row, or row where the first element is a prime number, all the numbers in that row (excluding the 1ís) are divisible by the prime.  For example, in the seventh row (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.  Yet, in a composite numbered row, such as row 6 (1 6 15 20 15 6 1), 15 and 20 are not divisible by 6.  In more mathematical terms it can be stated: ďif n is a prime number, then all the middle terms (all terms except the two end terms) of the nth row are divisible by n.  On the other hand, if n is a composite number, then some terms in the nth row will not be divisible by nĒ (Clawson 134).  This is yet another pattern found within Pascalís Triangle. 

     Several other patterns of Pascalís Triangle can be discovered by examining its diagonals. 

Fig. 3

The second diagonal, displayed in Fig. 3 as n is the sequence of counting numbers (1, 2, 3, 4, . . .).  Looking at the third diagonal, labeled tn in Fig. 3, one may spot the set of triangular numbers (1, 3, 6, 10, . . .).  Another sequence located in the triangle is the set of tetrahedral numbers, or the sums of the triangular numbers (1, 4, 10, 20, . . .).  These numbers are given in the fourth diagonal, labeled Tn in Fig. 3.  What fascinating discoveries! 

     Another pattern within the triangle is the Hockey Stick Pattern.  To understand the name of this pattern one should take a look at Fig. 4. 

Fig. 4

This pattern is as follows: the diagonal of numbers of any length starting with any of the 1ís bordering the sides of the triangle and ending on any number inside the triangle is equal to the number below the last number of the diagonal, which is not on the diagonal.  A few examples of this, also shown in Fig. 4,  are: 

                                             1 + 9                  =  10 
                                             1 + 5 + 15          =  21
                                             1 + 6 + 21 + 56  =  84.

The interesting Hockey Stick Pattern of Pascalís Triangle holds true for any set of numbers fitting the above definition. 

    The patterns that I have discussed so far involving powers of two, prime, counting, triangular, and tetrahedral numbers, and the hockey stick may seem to some to have been easily discovered.  The next pattern, that of Fibonnaciís sequence, is not always seen during oneís first glance of Pascalís Triangle.  Yet, as it may appear to one who examines a representation of the triangle that has several drawn-in slanted lines [see Fig. 5], there is indeed a connection between Pascalís Triangle and Fibonnaciís sequence. 

Fig. 5

Recall that Fibonacciís sequence begins with two 1ís and then all other numbers are generated by summing the two numbers before that term in the sequence. As Figure 5 reveals, the numbers located on each of the drawn-in diagonals of Pascalís Triangle sum to the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, . . .).  This is just one more fascinating internal mystery of the triangle.  I encourage you to discover for yourself other secrets and patterns in the numbers of Pascalís Triangle.