Lets expand a few with increasing values of n. What part do these numbers play in mathematics? Lets first look at the binomial expansion of (a + b)^n. N=4 in this case, since we started with n=0 at the top. The value of the number that goes in the rth place of that row isĬonsider, for example, the fifth row of the triangle that has elements 1 4 6 4 1 Let the number of the number in the row be r, starting from r=1 at the left. Let the row number start from 0 at the top. How can you determine the numbers in any row without knowing the numbers in the previous row? The 2nd and 3rd numbers in the 3rd row are the sum of the 1's and 2's above them. The last one is derived the same as the first 1 in the row. The next number is the sum of the two 1's above it.

Looking at the 2nd row, the 1 at the beginning of each row is the sum of the 1 and the implied 0 above it. In all its simplicity, each number is simply the sum of the two numbers immediately above. If you do not see the evolving pattern, now is probably the best time to explain how to create the rest of the triangle. The next row contains a 1, 3, 3, and 1 as in The next row contains a 1, 2, and 1 as in The apex of our triangle is the number 1 and called row 0. We are going to create a triangle with an array of numbers within the triangle. But lets first define the array and then show how it applies. The first appearances of the triangle were allegedly associated with the coefficients of the binomial expansion. Surprisingly, it has connections with probability, combinations, the binomial expansion, Taxicab Geometry, powers of 2, and perhaps many others which I have not yet had the privilege of hearing about. Blaise Pascal had the distinction of having it named after him merely because of his extensive 17th century work with it in relation to probability. Contrary to popular belief, it was not created by Pascal but is believed to have been created, or discovered, by both the Chinese and Persians sometime during the 11th and 12th centuries. There is one famous arrangement of numbers in a familiar geometric shape that has received the attention and admiration of mathematicians for centuries, Pascal's Triangle. You might say that the digits are parasitic after being operated on. For example 102,564 x 4 = 410,256 arrived at by merely moving the last 4 to the beginning of the remaining 5 digits. By definition, all pandigital numbers are divisible by nine.Ī parasite number is one that retains the exact same digits when multiplied or divided by another number. The smallest pandigital number is 1,023,456,789. There is a chapter on palindromes in Martin Gardner's book, Mathematical Circus, published by The Mathematical Association of America, 1992, pages 242 - 252.Ī pandigital number is one that contains each of the digits from zero to nine exactly once with the leading digit never being a zero. Have you ever heard of a Pandigital Prime? A Pandigital Prime is one that contains all 10 digits 0 through 9. You might like to seek out other palindromic primes or better still, how many palindromic primes can you find that are in arithmetic progression such as 13931, 14741, 1551 with a common difference of 810? Palindrome 868 and the number 89 leads to the palindrome 8,813,200,023,188. It has been said that every number will eventually produce a palindrome. Continue this process until the sum is palindromic, which might require many steps. Reverse the digits and add to the original number. If you have some spare time: Pick a number with any number of digits. Some palindromic cubes derive from the cubes of 7, 11, 101, 111, 1001, 10001, 10101, etc., which are themselves palindromic. Can you find a palindromic square number with an even number of digits? DON'T LOOK (836^2 = 698896 There is no smaller such number.) The number 26 is the smallest non palindromic number whose square happens to be palindromic, 676. There are of course others that do not follow the simplistic pattern of the numbers consisting totally of 1's such as the square of 22 which is 484, 264 which is 69696, or 2285 which is 5221225. The obvious squares are the squares of 1, 11, 111, 1111, etc. The regular numbers, such as the examples just mentioned, are infinite in quantity. Palindromic numbers can be prime, composite, odd, even, square, cube, and so on. A palindrome is a number that reads the same backwards as forwards, such as 121, 363, 3883, 37973, to mention a few.

Palindromic sequence math