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PUDUCHERRY U.T. SCHOOL TEACHERS’ FEDREATION
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Hello Mathematics lovers! Here are some Mathematical fascinating number
patterns, puzzles, magic squares. Enjoy the patterns and try to solve the magic
squares and puzzles. We will bring you more patterns like this in the forthcoming
days.
Compiled by:
G.THULASI, M.Sc., M.Ed., M.Phill., PGDCA.,
Trained Graduate Teacher,
Navalar Nedunchezhian Government Higher Secondary School,
Lawspet, Puducherry – 605 008.
Cell: +91- 95784 72719 , +91- 94428 91429
e-mail: [email protected]
Enjoy Math world…!!!!!!!!!!!!!!
*********
PUZZLE-1
A man walked from Puducherry to Chennai at a uniform speed. After walking 60 k.m. he
walked back to Puducherry increasing his speed by 1 k.m. per hour. He took 5 hours less for the
return journey . What was his speed during the forward journey ?
PUZZLE-2
Ramu sold a book for Rs.96, gaining as much percent as the book had cost Gopu. How
much did I pay for it?
PUZZLE-3
A labour earned Rs.150 in a certain number of days. If he had worked for five days
more, but opting for Rs.2.50 less for each day, he would have still earned the same amount.
How many days did he work ?
PUZZLE-4
A bird perched on a flag pole, 12 metres tall, spotted a insect 36 metres away from the
pole, running towards the pole. It instantly dived in the line of the hypotenuse and caught the
insect when the distances cove red by both were the same. At what distance from the pole did
the bird size the insect.
SOLUTION TO PUZZLES:
1. Let the man walk at x km/hr during the forward journey.
Now the time taken for the forward journey is Speed = Distance / time = 60/x
Similarly the time taken for the return journey is Speed = Distance / time = 60/ x + 1 hrs.
Now the difference is 60/ x - 60/x+1 = 5 Rewriting the equation we get
X2 + X – 12 = 0. It factors are (x-3)(x+4) = 0 gives x = 3 or x = -4
Therefore the speed is 3 km/hr.
****************************
2. Let the cost of the book be Rs. X. Then the profit is = x% or x/100.
Now x(100 + x)/ 100 = 96 Rewriting the equation we get
X2 + 100 X – 9600 = 0 Solving the equation (x-60)(x+160) = 0
Here we can not take negative value therefore x = 60 i.e. The cost of the book is Rs.60.
3. Let the number of days he worked be x. Then he got
Rs.150/x per day
Now (150/x – 2.50)(x + 5) = 150 Re writing the equation we have
X2 + 5X -300 = 0 Solving this (x-15)(x+20) = 0 Therefore x = 15 or x = -20
Taking the positive value The answer is 15 days.
4. The given problem can be represented as a Right angled triangle.
B
12
A
X
C
36-x
D
Let AB = x metres. Now DC = BC = 36-X
Also AB2 + AC2 = BC2 i.e. 122 + x2 = (36- x)2 Solving this equation we x = 16 metres.
WRITING NUMBERS IN EXPONENTIAL FORM
When you want to express either very large or very small numbers it is usual to
write them in this compact form.
1010
= 10,000,000,000
109
=
1,000,000,000
108
=
100,000,000
107
=
10,000,000
106
=
1,000,000
105
=
100,000
104
=
10,000
103
=
1,000
102
=
100
101
=
10
100
=
1
10-1
=
0.1
10-2
=
0.01
10-3
=
0.001
10-4
=
0.0001
10-5
=
0.00001
10-6
=
0.000001
10-7
=
0.0000001
10-8
=
0.00000001
10-9
=
0.000000001
10-10
=
0.0000000001
*************
OUT IN SIX
The object of this calculator game is to reach zero in just six goes, and you
won’t be competing with anyone else, because this is a game for you to play by
yourself. This is what you do:
1. Enter any six-digit number into your calculator, providing that you do not
use zero and that no digits is repeated.
2. You now have six goes to try and reach zero.
3. You can use and mathematical function you like in each go and you can use
any two-digit number.
4. Although it might seem obvious to keep dividing and subtracting to reduce
your total, you may find it better to add or even multiply to reach a number
that can then be cleanly divided by another.
(Once you’ve mastered bringing it down to zero in six goes, why not try
doing it in five?)
Example
1. Suppose you enter 583621
2. In your first move you add 19
Then you divide by 40
Then you subtract 11
Then you divide by 45
Then you divide by 12
Then you subtract 27 from the last go
NOTE: Try like the above.
************
=
=
=
=
=
583621
+
19
583640
14591
14580
324
27
0
TWO’S COMPANY
2 x 5 = 10 doesn’t it? Well, by repeating the number tow five times it’s possible to
produce the value of all the digits from zero to nine, or in this case form one to
zero.
Watch:
2 + 2 – 2 – 2/2
=1
2+2+2–2-2
=2
2 + 2 – 2 + 2/2
=3
2x2x2–2-2
=4
2 + 2 + 2 - 2/2
=5
2+2+2+2-2
=6
22/2 – 2 - 2
=7
2x2x2+2-2
=8
2 x 2 x 2 + 2/2
=9
2 – 2/2 - 2/2
=0
*************
SUMS OF THE CENTURY
I used to play Scrabble with a centenarian. He was called John Badley and
he founded Bedales School in Hampshire where I was given (well, offered) my
education. This is just the sort of mental recreation Mr Badley enjoyed I his
eleventh decade: simple sums that only use the nine digits once and in their
correct order and only require multiplication, addition or subtraction to come
up with the same answer every time: 100
1 + 2 + 3 + 4 + 5 + 6 + 7 + (8 x 9)
= 100
- (1 x 2) – 3 – 4 – 5 + (6 x 7) + (8 x 9) = 100
1 + (2 x 3)+(4 x 5) - 6 + 7 + (8 x 9)
= 100
(1 + 2 - 3- 4)( 5 - 6 - 7 - 8 – 9)
= 100
1 + (2 x 3) + 4 + 5 + 67 + 8 + 9
= 100
(1 x 2) + 34 + 56 + 7 - 8 + 9
= 100
12 + 3 - 4 + 5 + 67 + 8 + 9
= 100
123 – 4 –5 – 6 - 7 + 8 - 9
= 100
***********
ALL THE NINES
All the digits from one to nine are used in these multiplication sums, all that is
except for eight. However, if you multiply 123456789 by the first nine and if the
digit in each product is multiplied by the number of digits in each product (nine)
you’ll find that another interesting product is produced.
123456789 x 9 = 111111111
123456789 x 18 = 2222222222
123456789 x 27 = 3333333333
123456789 x 36 = 4444444444
123456789 x 45 = 5555555555
123456789 x 54 = 6666666666
123456789 x 63 = 7777777777
123456789 x 72 = 8888888888
123456789 x 81 = 9999999999
***********
THE MISSING NUMBERS
There’s a number missing in each of these puzzles. Can you work out what each
one should be?
a)
b)
9
11
4
2
6
60
32
80
1
2
82
6
0
4
3
2
1
5
c)
63
94
79
54
81
45
72
54
36
63
45
d)
632
36
981
72
463
72
333
e)
f)
4
704
0
212
4
16
6
1
30
12
9
11
9
3
6
22
4
4
2
94
62
76
11
83
13
23
53
10
************
A Primer on Prime Numbers
Hello friends ! Look at the number 221 . Look at the fascinating sense in this number .
Actually 221 is a composite number which is the product of two prime numbers 13 and 17.The
number 221 also has other attractive features. It’s also the sum of five and nine consecutive
prime numbers i.e. (37 + 41 + 43 +47 + 53 = 221) and (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41
= 221).
What is a Prime number ?
A prime number, or a prime, is an integer greater than 1 that can be divided only by itself and 1.
A natural number greater than 1 that is not a prime number is called a composite number. For
example, 5 is a prime, as it is divisible by only 1 and 5, whereas 6 is composite, because it has
the divisors 2 and 3 in addition to 1 and 6. ‘Are there any limitation to prime numbers in
number world?’
No. In fact, there are an infinite number of primes. Another way of stating this is that the
sequence 2, 3, 5, 7, 11,13, ... of prime numbers never ends. Most of the unsolved mysteries in
mathematics are also related to prime numbers.Now can we say the number 1 a prime?’ No ‘1
is not a prime number. 2 is the first prime number and the only even prime number; all other
prime numbers are odd,’o.k.
Is zero a prime?’ then also the answer is No.It is interesting to know that zero is neither a prime
nor a composite number. It cannot be a prime because it has an infinite number of factors. It is
not a composite number because it cannot be expressed by multiplying prime numbers. 0 must
always be one of the factors.’ ‘Are all composite numbers formed by multiplying primes?’
Yes. Let me explain. If we factorize a composite number into two smaller numbers, then it
needs to be checked whether these two numbers are themselves primes or composites. For
example, 6 factorizes into 2 x 3. Both the numbers 2 and 3 are prime numbers. The number 18
factorizes into 2 x 9. Here the number 2 is a prime but the number 9 is not. However, the
number 9 factorizes into 3 x 3 and the number 3 is a prime. Hence the number 18 can be
written as 18 = 2 x 3 x 3. Any composite number, no matter however large, can be factorised
into two smaller numbers. We then ask whether each of the smaller factors is a prime or
composite. If either one is composite, we factorize it again. The process continues till all the
factors are primes. This in itself is interesting and leads to a fascinating conclusion. When a
composite number is factorized into primes, those primes are unique to that number.
For example, we can factorize the number 30 into 2 x 3 x 5. No other set of primes, when
multiplied together, will yield 30.’ ‘This is very interesting,‘This interesting fact leads to one of
the building blocks of mathematics, viz., every whole number greater than 1 can be expressed
as a product of prime numbers in one and only one way, which has come to be known as the
fundamental theory of arithmetic.’ Now we can understand why the number zero cannot be a
composite number. The number zero can be expressed as 0 = 0 x 2 x 3 or 0 = 0 x 7 x 17 or
infinitely many different ways. A composite number can be expressed as a product of prime
numbers in one and only one way. Hence the number zero cannot be a composite number. At
the same time zero has infinite numbers of factors. Hence it cannot be a prime.
If 1 is considered to be a prime, then the fundamental theory of arithmetic breaks down!
Because 30 = 2 x 3 x 5 and also 30 = 1 x 2 x 3 x 5. Hence factorization of 30 will not be unique.
Therefore, 1 is not a prime number,’ ‘Prime numbers are randomly distributed among the
natural numbers, without any apparent pattern. However, the global distribution of primes
reveals a remarkably smooth regularity. If you arrange all positive integers in a table having six
columns, then you will notice that all the primes, except 2 and 3, are either in column 1 or
column 5. You can try to expand the table and you will note that all prime numbers, except 2
and 3, will appear either in column 1 or in column 5.
However, the distribution of primes within column 1 and 5 is random,’ ‘Yes. After the Greeks,
major documentary evidence of the study of prime numbers appeared in the 17th century. In
1640, French lawyer and an amateur mathematician Pierre de Fermat, while researching
perfect numbers came up with a formula to generate prime numbers. Fermat conjectured that
all numbers of the form ( 22n+1), where n is any natural number, are prime. Fermat verified this
up to n = 4 (or 216 + 1). Prime numbers generated from this equation are known as Fermat
number. Fermat could not verify it beyond n = 4. Later it was found that the very next Fermat
number 232 + 1 is a composite, 641 being one of its prime factors. In fact, no further Fermat
number is known to be a prime.’‘‘Prime numbers are not uniformly distributed. For example,
between 1 and 100 there are 25 primes, while there are 21 primes between 101 to 200.’
Does the frequency of prime numbers reduce as we move towards larger number group?”
‘Yes, that too with some regular pattern. German mathematician and physical scientist
Friedrich Gauss did significant work on prime numbers. Gauss spent hours trying to figure out
some pattern or regularity in the distribution Pierre de Fermat (1601- 1665) of prime numbers.
Initially he confirmed the findings of the ancient Greeks that there appeared to be no pattern.
However, later he discovered that if numbers are grouped according to powers of 10 (that is: 110, 1-100, 1-1000, etc) and then if one picks a number at random from within each range, the
probability of it being a prime has some regular pattern.’ ‘It is fascinating to see the
mathematical pattern in apparent randomness.’ ‘True. In this way regularity appeared out of
the mist of disorder. Each time a larger number group was considered, the probability of
getting a prime number went down. That is, as the numbers got bigger, the prime numbers
thinned out according to a predictable pattern. This eventually led to the prime number
theorem (PNT) that describes the asymptotic distribution of the prime numbers. The prime
number theorem gives a general description of how the primes are distributed amongst the
positive integers. The prime number theorem states that the number of primes less than n is
approximately n divided by the logarithm of n.’
Let us have a simple formula to obtain a prime number in a sequence?’
Despite efforts by all leading mathematicians, there is no formula for computing the nth prime.
Many formulae do exist that produce nothing but prime numbers. However, these formulae do
not produce each successive prime nor do they predict the next prime in sequence.’ ‘What is
the largest known prime number?’ ‘Till now, the largest known prime is 243112609 – 1. It is a
Mersenne prime. French philosopher and mathematician Marin Mersenne showed that all
numbers in the form (2p – 1) are prime numbers, where ‘p’ is a prime. The largest known prime
has almost always been a Mersenne prime.’Based on the difficulty of factorizing a product of
two very large primes, public-key cryptography was invented. Public key cryptography
algorithms utilize prime numbers extensively. Prime factorization is the key to all e-commerce
applications, where financial transactions are done over Internet,’
What’s the special thing about the number 221?’ I asked. ‘In mathematics, a semiprime (also
called biprime or 2-almost prime, or pq number) is a natural number that is the product of two
(not necessarily distinct) prime numbers. As I mentioned before, 221 is the product of two
prime numbers, 13 and 17. So,221 is a semiprime number. Examples of a few other semiprime
numbers are: 4, 6, 9, 10, 14, 15, 21, 22, 25 and 26.’ ‘That’s very fascinating indeed. What are the
characteristics of the semiprime numbers?’ ‘The square of any prime number is a semiprime, so
the largest known semiprime will always be the square of the A Primer on Prime Numbers
Moments in Mathematics largest known prime, unless the factors of the semiprimes are not
known. As you could guess, based on the Mersenne prime, the largest known semiprime is
(243112609 - 1)2, which has over 25 million digits. Like prime numbers, the semiprime numbers
are also very important for cryptography and number theory.’
Source:: Moments in Mathematics by Rintu Nath.