The document discusses the history of mathematics and various patterns in numbers such as magic squares, magic stars, triangular numbers, Fibonacci numbers, and Pascal's triangle. It provides examples and properties of each type of pattern. For instance, it explains that a magic square of order 8 forms another magic square if columns are rearranged and that triangular numbers represent the number of dots that can form an equilateral triangle.

2. THE 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.)

3. PANDIAGONAL MAGIC SQUARE
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
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.

4. MAGIC STARS
Magic 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

5. 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)/2
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.
A Triangular number can never end in 2,
4, 7 or 9.
All perfect numbers are triangular
numbers
The only triangular number which is
prime is 3.

6. • 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.

8. 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.

9. 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.

10. 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.

11. CONNECTION 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 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).

12. 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 ...
•

13. 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.

14. Ronik
Sudiksha Tavishi a
Patterns in numbers