THE HISTORYTHE HISTORY

The history of mathematics is a history of people fascinated by numbers. A driving force in mathematical development has always been the need to solve practical problems. However, man's innate curiosity and love of pattern has probably had an equal part in its development. Most written records of early mathematics that have survived to modern times were actually lists of mathematical problems i.e. recreational mathematics. Examples: the Rhind Papyrus, (circa  1700 BC), a series of 87 problems, was the key to deciphering Egyptian hieroglyphs; Diophantus' Arithmetica (circa 250 BC), a collection of 130 mathematical problems with numerical solutions of determinate equations. (Fermat's Last Theorem was found written in the margin of a copy of this book.)

PANDIAGONAL MAGIC SQUAREPANDIAGONAL MAGIC SQUARE

This order 8 magic square has the interesting property that alternating numbers in each row, column, and the main diagonals sum to 130. Each quarter (layer of the magic cube) is itself a magic square. The cube is pandiagonal between layers. It is not pandiagonal within each layer because NO order 4 cube can be perfectly magic AND pandiagonal in 3 dimensions.

A of type of magic square: used to describe a magic square that forms another magic square if any number of columns are taken as a unit from one side and put on the other

MAGIC STARSMAGIC STARSMagic stars are similar to Magic Squares in many ways. The order refers to the number of points in the pattern. A standard (normal or pure) magic star always contains 4 numbers in each line and consists of the series from 1 to 2n where n is the order of the star. The magic sum (S) equals (Sum of the series/number of points) plus 2 or S = 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

TRIANGULAR NUMBER SEQUENCE 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.

Flocks of birds often fly in this triangular formation. Even several airplanes when flying together constitute this formation. The properties of such numbers were first studied by ancient Greek mathematicians, particularly the Pythagoreans.

Rule: xn = n(n+1)/2

A Triangular number can never end in 2, 4, 7 or 9.

All perfect numbers are triangular numbersThe only triangular number which is prime is 3.

• Palindromic Triangular Numbers: 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: 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.

APPLICATIONS OF PASCAL’S TRIANGLEAPPLICATIONS 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 PATTERNHOCKEY 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. 

A few examples of this, also shown 1 + 9=10                                                                 1+ 5 + 15=21                                                       1+ 6 + 21 + 56 =84

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

PASCAL’S TRIANGLEPASCAL’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……,nWhen 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  =  2 1                                            1 + 2 + 1             =   4  =  2 2                                            1 + 3 + 3 + 1       =   8  =  2 3                                            1 + 4 + 6 + 4 + 1 = 16  =  24.

CONNECTION TO SIERPINSKI’S TRIANGLECONNECTION TO SIERPINSKI’S TRIANGLE

• Sierpinski's 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 ConstructionThe 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).

FIBONACCIFIBONACCI NUMBERSNUMBERS

• 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 ...

•   

One of the most fascinating things about the Fibonacci numbers is their connection to nature. Some items in nature that are connected to the Fibonacci numbers are:

- the growth of buds on trees

- the pinecone's rows

- the sandollar

- the starfish

- the petals on various flowers such as the cosmos, iris, buttercup, daisy, and the sunflower

- the appendages and chambers on many fruits and vegetables such as the lemon, apple, chile, and the artichoke.

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Patterns in Patterns in numbersnumbers

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