Patterns in numbers

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Stress your brain to search the innumerable patterns hidden in numbers which we usually overlook in Maths.From Sierpinski Triangle to Pandiagonal Magic Square, Fibonnaci Numbers to Hockey Stick …

Stress your brain to search the innumerable patterns hidden in numbers which we usually overlook in Maths.From Sierpinski Triangle to Pandiagonal Magic Square, Fibonnaci Numbers to Hockey Stick Pattern.Explore how they are used in daily life, in espionage.....?? technology and glare at the inventories.

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  • 1. THE HISTORYThe history of mathematics is a history of people fascinated by numbers. Adriving force in mathematical development has always been the need to solvepractical problems. However, mans innate curiosity and love of pattern hasprobably had an equal part in its development. Most written records of earlymathematics that have survived to modern times were actually lists ofmathematical problems i.e. recreational mathematics. Examples: the RhindPapyrus, (circa  1700 BC), a series of 87 problems, was the key todeciphering Egyptian hieroglyphs; Diophantus Arithmetica (circa 250 BC), acollection of 130 mathematical problems with numerical solutions ofdeterminate equations. (Fermats Last Theorem was found written in themargin of a copy of this book.)
  • 2. PANDIAGONAL MAGIC SQUAREA of type of magic square: used todescribe a magic square that forms anothermagic square if any number of columnsare taken as a unit from one side and puton the otherThis order 8 magic square has theinteresting property that alternatingnumbers in each row, column, and themain diagonals sum to 130. Each quarter(layer of the magic cube) is itself a magicsquare. The cube is pandiagonal betweenlayers. It is not pandiagonal within eachlayer because NO order 4 cube can beperfectly magic AND pandiagonal in 3dimensions.
  • 3. MAGIC STARSMagic stars are similar to Magic Squares in many ways. The order refers tothe number of points in the pattern. A standard (normal or pure) magic staralways contains 4 numbers in each line and consists of the series from 1 to2n where n is the order of the star.The magic sum (S) equals(Sum of the series/number of points) plus 2 orS = 4n + 2 This particular pattern (the only one of the 12) has numbers 1 to 5 at the points. Order-5 is the smallest possible magic star. However, it is not a pure magic star because it cannot be formed with the 10 consecutive numbers from 1 to 10. The lowest possible magic sum (24) is formed with the numbers from 1 to 12, leaving out the 7 and the 11. It is also possible to form 12 basic solutions with the constant 28, by leaving out the 2 and the 6
  • 4. TRIANGULAR NUMBER SEQUENCE A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n. Rule: xn = n(n+1)/2Flocks of birds often fly in this triangular formation. Even several airplaneswhen flying together constitute this formation. The properties of suchnumbers were first studied by ancient Greek mathematicians, particularlythe Pythagoreans. A Triangular number can never end in 2, 4, 7 or 9.All perfect numbers are triangularnumbersThe only triangular number which isprime is 3.
  • 5. • Palindromic Triangular Numbers: Some of the many triangular numbers, which are also palindromic ( i.e. reading the same forward as well as backward) are 1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15051, 66066, 617716, 828828, 1269621, 1680861, 3544453, 5073705, 5676765, 6295926, 351335153, 61477416, 178727871, 1264114621, 1634004361 etc. These can be termed as palindromic triangular numbers. There are 28 Palindromic Triangular numbers below 1010.• Square Triangular Numbers: There are infinitely many triangular numbers, which are also squares as given by the series 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 etc. These can be termed as Square triangular(ST) numbers.• The only Fibonacci Numbers that are also triangular are 1, 3, 21 and 55.
  • 6. APPLICATIONS OF PASCAL’S TRIANGLE• It can be used in real life for simplest things such as counting the number of paths or routes between two points.• It is used to count the different paths that water overflowing from the top bucket could take to each of the buckets in the bottom row. The water has one path to each of the buckets in the second row. There is one path to each outer bucket of the third row but two paths to the middle bucket and so on. HOCKEY STICK PATTERN Another pattern within the triangle is the Hockey Stick Pattern 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.
  • 7. A few examples of this, also shown 1 + 9=10 1+ 5 + 15=21 1+ 6 + 21 + 56 =84The interesting Hockey Stick Pattern of Pascal’s Triangle holds true for anyset of numbers fitting the above definition.
  • 8. PASCAL’S TRIANGLE• Pascal’s Triangle is named after Blaise Pascal who was a French mathematician, physicist and religious philosopher. With the help of this Triangle Pascal was able to solve the problems in probability.• It is an arrangement of binomial coefficients in a Triangular array known as Pascal’s Triangle.• The nth row in the triangle consists of binomial coefficients.• {N}• {k}, k =0,1……,n When two adjacent binomial coefficients in this triangle are added, the• binomial coefficient in the next row between them is produced. 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 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 = 2 4.
  • 9. CONNECTION TO SIERPINSKI’S TRIANGLE• Sierpinskis Triangle is at the same time one of the most interesting and one of the simplest fractal shapes in existence. A fractal is a geometric construction that is self-similar at different scales.• Geometric Construction The most conceptually simple way of generating the Sierpinski Triangle is to begin with a (usually, but not necessarily, equilateral) triangle (first figure below). Connect the midpoints of each side to form four separate triangles, and cut out the triangle in the center (second figure). For each of the three remaining triangles, perform this same act (third figure). Iterate infinitely (final figure).
  • 10. FIBONACCI NUMBERS• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …• Each term in the Fibonacci sequence is called a Fibonacci number. As can be seen from the Fibonacci sequence, each Fibonacci number is obtained by adding the two previous Fibonacci numbers together. For example, the next Fibonacci number can be obtained by adding 144 and 89. Thus, the next Fibonacci number is 233.• The Rule is xn = xn-1 + xn-2• where:• xn is term number "n"• xn-1 is the previous term (n-1)• xn-2 is the term before that (n-2)• The terms are numbered form 0 onwards like this:• n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...• xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ...•  
  • 11. One of the most fascinating things about the Fibonacci numbers is theirconnection to nature. Some items in nature that are connected to the Fibonaccinumbers are:- the growth of buds on trees- the pinecones rows- the sandollar- the starfish- the petals on various flowers such as the cosmos, iris, buttercup, daisy, and thesunflower- the appendages and chambers on many fruits and vegetables such as the lemon,apple, chile, and the artichoke.
  • 12. RonikSudiksha Tavishi a Patterns in numbers