Enjoying Fun in Learning Mathematics. Not an in-depth mathematics lecture but a few tips and tricks only.

1. LEARNING CAN BE FUN

2. Only Human can be Educated, others can
be Trained only.
• “The play of the human mind is the world’s most
exciting adventure”.
• “It’s very simple, It’s the three Cs: Character, culture
and Curriculum” [Vivekananda].
• “Not from one mind to another but from one heart
to another heart.” [Jogajog vs. Sangjog]
• “Kabil Bano, Kamyabi apne aap pichhe pichhe aa
jayega” [3 Idiots]

3. “There is no branch of mathematics, however abstract,
which may not some day be applied to phenomena of
the real world”
Nikolai Lobatchevsky
“A man is like a fraction whose numerator is what he is
and whose denominator is what he thinks of himself.
The larger the denominator, smaller is the fraction”
Tolstoy
“A mathematician is a blind man in a dark room
looking for a black cat which isn’t there”
Charles Darwin.
312/28/2018

4. Playing with Words
• What has four(4) letters, sometimes nine (9) letters, but never has
five (5) letters.
 TRUE.
• One word fills all 4 blank spaces.
A rich man needs ----------
A poor man has ----------
If you eat ------- you die and when you die
you can take ----------- with you!
 NOTHING
• 100 words without “a”.
 One, two, three, four, …… 999

5. Playing with Words
Dec. 25 = Oct. 31
123 = 1x100 + 2x10 + 3x1
= 1X102 + 2x101 + 3x100 = 12310
= 1x82 + 2x81 + 3x80 = 838
318 = 3x81 + 1x80 = 24+1 = 2510

6. a + b > c? or,
or, shortest distance?
• Shortest distance
between two points?

7. From Syllabus
Divisibility Test of S by p
• By 2: the last digit is even (2, 4, 6, 8, or 0)
• By 5: the last digit is either 0 or 5
• By 3: sum of all the digits is divisible by 3
By 7?
• Drop the last digit (a)from S; S N
• subtract two times that last digit from N
• N – 2x a = C (say); if C is divisible by 7, so is S.
• S=154, N=15, a = 4, C=N – 2x a = 15 - 2x4=7;
• both 7 and hence 154 are divisible by 7.

8. From Syllabus
Divisibility Test of S by p
By 11?
Example:
 S = 154, C = (4+1) – 5= 0. C= 0. So 154, is divisible.
 S = 4785, C = (4+8) –(5+7) = 12 – 12 = 0
 S = 9570, C = (9+7) – (5+0) = 16 – 5 =11

9. Funny Division (Howlers)
• 26/65 = 2/5
• 16/64 = 1/4
• 19/95 = 1/5
• (xy)/(yz) = x/z
• (10x + y)z = (10y + z)x, or
• Z = 10yx/(9x+y)

10. Thinking Out of Box
• Use 6 matches (of equal length)
• to make 4 equilateral triangles
• (no broken stick)

11. X

13. Connect all the 9 points by Only 4 straight lines,
without lifting the pencil, without overtracing

14. 1
2
34

15. Pair up and dance. Pair-wise summation.
1 +2 + 3 + 4 + …98+ 99 + 100 =?

16. 1 +2 + 3 + 4 + …98+ 99 + 100 =?
101 101 101
50 pairs of 101 = 50 x 101
= 50X100 + 50
= 5000 + 50
=5050

17. 1 = 2 ???
E Ki Re Baba !!! (Paradox-1)
• Let a =b
a2 = ab
a2 – b2 = ab – b2
(a+b)(a-b) = b(a-b)
a+b = b
2b = b
2 = 1

18. -1 is (+)ve !!!
E Ki Re Baba !!! (Paradox-2)
• Let a =1+2+4+8+16+32+…; a>0, (+)ve
 a -1 = 2+4+8+16+32 + …
 2x a = 2+4+8+16+32+…
 2x a = a -1, or
 2a –a =-1
 a = -1, but a>0, (+)ve
 -1 >0

19. 1 foot = 144 inches !!!
E Ki Re Baba !!! (Paradox-3)
• 1 foot = 12 inches
• 2 feet = 24 inches
• (1/2) feet = 6 inches
• Now multiplying
• 2 x (1/2) feet = 24 x 6 inches = 144 inches
• 1 foot = 144 inches
 1 square foot = 144 square inches.

20. -1 = +1 !!!
E Ki Re Baba !!! (Paradox-4)
• √-1 x √-1 = √(-1 x -1) = √1 = 1
• √-1 x √-1 = (√-1)2 = -1
• Equating two equations
• 1 = -1

21. 1 = 0 !!!
E Ki Re Baba !!! (Paradox-5)
S = 1 -1 +1 -1 +1 -1 +1 -1 + …
S =(1-1)+(1-1)+(1-1)+(1 -1)+ …
S = 0+0+0+0+0 + …
S=0
S = 1 -1 +1 -1 +1 -1 +1 -1 + …
S = 1– (1-1)-(1-1)-(1-1)-(1-1)- …
S = 1-0-0-0-0-0- …
S = 1
Therefore, 0 = 1

22. Resolve unsolvability
bridges of Konisberg [Euler 1736]
The first problem in Graph Theory

23. Graphical
1+ 1/2 + 1 /4 +1 /8 + … = ?
1/4
1
1
2312/28/2018

24. Graphical
1/2
1/4
1
1/2
2412/28/2018

25. Graphical
1/4
1/4
1
1/2
2512/28/2018

26. Graphical
1/8
1/4
1/8
1
1/2
2612/28/2018

27. Graphical
1/16
1/4
1/8
1
1/2
2712/28/2018

28. Graphical
1/16 1/32
1/32
1/4
1/8
1
1/2
2812/28/2018

29. Play with Tricks/Logic-1
• Draw a circle around a person so that he will not be
able to jump out of it, though his legs are free.
• Stretch your hands apart, having a coin in each hand.
Without bringing your hands together, cause both to
come into the same hand.
• You have two clocks: one does not go at all, and the
other loses a minute a day; which would you prefer?
• The day before yesterday I was 16 years old. Next
year I shall be old enough to cast my first vote.
 DoB: 31 December, Today is January 1

30. Play with Tricks/Logic-2
• Here are 10 numbered statements. How many of them are
True?
• 1. exactly one of these statements is false.
• 2. exactly two of these statements are false.
• 3. exactly three of these statements are false.
• 4. exactly four of these statements are false.
• 5. exactly five of these statements are false.
• 6. exactly six of these statements are false.
• 7. exactly seven of these statements are false.
• 8. exactly eight of these statements are false.
• 9. exactly nine of these statements are false.
• 10. exactly ten of these statements are false.

31. Play with Tricks/Logic-3
To lose to win
• If you are asked to bet one rupee, that if you are
given 2 rupees, you will give 3 rupees in return.
• Will you accept the bet?
 If you want to win the bet, you are not a loser!
 If you lose the bet, you win 1 rupee.
• Each of you and a co-passenger gave a total of Rs.
10/- to the conductor. He asked you about your
destination but not the other one. Why?

32. Play with Tricks/Logic-4
• Show, by a simple example, that an irrational number
raised to an irrational power need not be irrational
number.
• Two legs sat on three legs when along came four legs and
stole the one leg, whereupon two legs picked up three
legs and threw at four legs, and got his one back. Explain
please.
• 3025  30 & 25  30+25 = 55, 552 = 3025;
• Check with 9801.
• “This sentance contanes one misteak”.
• How many mistakes are there?

33. Play Time-1,2,3
1. Mental Sum (rounding off error!)
2. Rubber band changing fingers
3. To put a knot on a string

34. Let’s Play a little more-4,5
• 12 coins; 5 H up, 7 T up.
• Make two groups taking and flicking as many as you like,
so that
• Numbers of H up coins are same in both the groups.
• Rule of Nine: 9 options would decide your “luck”.
• Take n (from 1 to 9), multiply by 3
• Add 3 to result
• Multiply result by 3.
• Add the digits.
• You have drawn your lucky option!
3x(3xN + 3)
= 9N + 9
= 10 N +(9 – N)
Sum of digits
= N+9-N = 9, always!

35. Let’s Play a little more-6
Who can reach 50 faster!
• You choose a number less than 11.
• Opponent adds any number ( < 11) to your number.
• Now you add any number (< 11) to his/her number.
• Now he/she adds …
• Then you add …
• That who reaches first 50
or more is the winner.
Winning strategy
6, 17, 28, 39
For 100
1, 12, 23, 34, 45, 56, 67, 78,
89

36. Let’s Play a little more-7
sum = product!
• There are two numbers whose sum equals their
product. They are 2 and 2. [X2 = 2X, or X = 2]
• Now, if I write any number N,
• Can you give me another number that would give a
like result by addition or multiplication?
• N and N/(N-1) are that friendly.
N x N/(N-1) = N2/(N-1)
N + N/(N-1) = N[1+1/(N-1)] = N[(N-1+1)/(N-1) ]
= N2/(N-1) [1x2x3 = 1+2+3 = 6 ]

37. Take Home Puzzles -1
Valentine Pair of Numbers
• M and N; a pair of friends
• Sum of factors of M = N
• Sum of factors of N = M
M=284, N=220
Sum of factors of 220 (N)
1+2+4+5+10+11+20+22+44+55+110= 284 = M
Sum of factors of 284 (M)
1+2+4+71+142= 220 = N
M = 36, sum of factors of 36
= 1+2+3+4+6+9+12+18= 55
Sum of factors of 55
=1+5+11= 17 ≠ 36
So 36 and 55 are not friendly.
 Check the pair
1184 & 1210

38. Take Home Puzzles -2
walking down the river
• Point S  river (P)  point T;
• Walk as short as possible!
T
S
P P

39. Take Home Puzzles
walking down the river
• Point S  river (P)  point T;
• Walk as short as possible!
T
S
P
R

40. Take Home Puzzles -3
more milk in water or more water in milk?
• One glass contains 100 ml milk and another glass
contains 100 ml water. A spoonful of milk is
transferred to the water, and the mixture is stirred.
Then, a spoonful of this mixture is transferred back
to the glass of milk.
Is there more milk in the water or more water in the
milk?

41. Take Home Puzzles -4
Is One Weighing Enough!
• 10 boxes of coins. Each box contains good coins,
weighing 1 g each except one box whose coins are
bad and weigh 2g each.
• How many weighing's are needed to decide which
box has the bad coins.
1+2+3+4+5+6+7+8+9+10 = 55  if all were good.
W – 55 = n, is the number of the bad-coins-box.
 56 – 55 = 1  first box; 60 – 55 = 5  fifth box, etc.

42. Take Home Puzzles -5
Kaprekar’s number
• Take any 3 digit number, arrange the digits in
ascending and descending order to form two
numbers, and subtract the smaller from the larger.
Repeat the same process with the answer.
• What is the result-eventually?
• 459 [for 4 digit number, it is 6174].

43. Take Home Puzzles -6
• In the multiplication problem below, each letter
represents a different digit:
A B C D E
x F
G G G G G
• Which of the ten digits does G represent?
• 9 5 2 3 8 x 7 = 6 6 6 6 6 6

44. Take Home Puzzles -7
• Tun and Taan were sharing their sandwiches; Tun had
5 and Taan had 3 sandwiches. You joined them
without any and offered 2 rupees for your part, of
which Tun took Rs. 1.25 and Taan took Rs. 0.75.
• You told them: the distribution is wrong!
• Are you right?
• Tun: 5 - 8/3 = 7/3, Taan: 3 – 8/3 = 1/3;
• So, ratio should be 7:1;
• Tun: Rs. 1.75, Taan: Rs. 0.25

45. Take Home Puzzles -8
How many squares on an 8x8 chessboard?
82 + 72 + 62 + 52 + 42 +32 + 22 + 12 = 204

46. What do you See ?
Lion, Elephant, Pig, Giraffe, Horse, Dog, Bear, Camel