Magic of NumbersMagic of Numbers

9474=94+44+74+44

548834=56+46

+86+86

+36+46

Playing with NumbersFeynman and the Abacus

Raios cubicos!

Numbers are toys.They are meant

to be played with, not feared of!!

You must know them!!

Game 1 : Figurate numbers

You see?Patterns!

Triangular Numbers

Formulaaaaa!!!!!!!!

n(n+1)

2

Pascal's Triangle

There are more ways!!

Moessner's magic

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Squares

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 4 9 16 25 36 ............ and so on

For cubes???

Moessner's magic

Triangular numbers1 2 3 4 5 61 3 6 10 15 21

Factorials

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 6 11 18 26 35 46 58 71 85

6 24 50 96 154 225 24 120 274

Patterns they just don't stop

Counting numbers

and Hexagonal numbers

Its him.Fermat!!

Game 2 : Representing Numbers

● Numerous ways to represent a single number

● e.g. 1729

– 1729 = = ● Particularly interesting in cases like these

– 153 = – 4150 =

103+93 123+13

13+53

+33

45+15

+55+05

A new puzzle;Time to trouble

friends!!

Number with same digits

● 24 = 8 + 8 + 8 = 22 + 2 ; so far so good

● 24 =

– Now we are talking● What else can you think of ?

33−3

There is exactly one more way

for 3 digits

24 puzzle

● 24 = (11 + 1)(1 + 1) = 3(3 x 3 – 3 / 3)

= 4(4 + 4) – (4 + 4) = (6+6)(6+6)/6

=

● Can we generalize it?

22+22/2

Magical 24

● Solution for 7 digits

e.g.

nn+nn+n+nn

77+77+7+77

=24 It is not the only way though.

Game 3 : Divisibility

● Divisibility tests

● Is there a test for every number?

● Are there more than one tests for a number?

3 – not limited to sum of digits

● Case of 283524 (2 + 8 + 3 + 5 + 2 + 4 = 24)

● Summing 2 digits at a time

– 28 + 35 + 24 = 87● Summing 3 digits at a time

– 283 + 524 = 807

What difference it makes?

● 9801 = 99 x 99 | 98 + 01 = 99

● Adding 2 digits at a time is test of 99

● What can be test for 999 then?

● Works for 3, 33, 333 etc. as well

Puzzling 111

● Test of 11 is not too different

– Subtraction is a kind of addition (-ve integers)● Will similar generalization work for 11, 111 etc. ?

● 234543 = 111 x 2113

● 23|45|43 23-45+43 = 21; not working

What are we missing?

● 9 = , 11 =

● 99 =

● 111 does not fit, 101 fits. Let's check!

● 596102 = 101 x 5902

● 59|61|02 59-61+02 = 0

101−1 101+1

102−1

Curious case of divisibility by 3

● 13|2 13 – 2 x 2 = 9

● Seems interesting; check for others

● 8805

– 880|5– 880 – 5 x 2 = 870– 87|0– 87 = 3 x 29

Curious case of divisibility by 3ददततीय

● 13|2 13 + 2 x 4 = 21

● 8805

– 880|5– 880 + 5 x 4 = 900

7 and 13

● 182 = 7 x 26

– 18|2 18 – 2 x 2 = 14

● 5512 = 13 x 424

– 551|2 551 + 2 x 4 = 559– 55|9 55 + 9 x 4 = 91

● Prime check

● Divisibility of 49, 51

● Fibonacci numbers

● Number of factorsFew more to think

by yourself