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Now knowing how to construct Pascal’s Triangle, one may wish to look deeper into its rows and diagonals to spot some of its greatest mysteries. One of the triangle’s 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 triangle’s numbers, patterns such as the powers of two, the powers of eleven, the triangular numbers, and Fibonnaci’s 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 Pascal’s 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 Pascal’s 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 2^{n}, when n is the number of the row. For example:
1
= 1 = 2^{0}
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 1’s) 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: “if 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 n” (Clawson 134). This is yet another pattern found within Pascal’s Triangle. Several other patterns of Pascal’s 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 t_{n} 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 T_{n} 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 1’s 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
The interesting Hockey Stick Pattern of Pascal’s 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 Fibonnaci’s sequence, is not always seen during one’s first glance of Pascal’s Triangle. Yet, as it may appear to one who examines a representation of the triangle that has several drawnin slanted lines [see Fig. 5], there is indeed a connection between Pascal’s Triangle and Fibonnaci’s sequence. Fig. 5 Recall that Fibonacci’s sequence begins with two 1’s 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 drawnin diagonals of Pascal’s 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 Pascal’s Triangle. 