Thought Experiment

• Take a piece of paper one thousandth of an inch thick

• Now fold it in half and then in half again

• Do this 50 times (I know that this is not practicable)

• How thick do you think it will be ?

• Each time we fold the paper, it doubles in thickness• So after the first fold it is 2x as thick• After the second fold it is 2x2 ie 4 times as thick• After 50 folds it is 2x2x2 .......... 2x2 times as thick

• That is 250

• Divide by a thousand to get the thickness in inches• Divide by 63,360 to get the thickness in miles

• And we get ......

250 = 1125899906842624

Dividing by a thousand = 1125899906842 inches

Dividing by 63360 = 17,769,885 miles

Carol recently gave a talk on the metal

Bismuth.

One of its interesting facts was it's half-life

This is currently calculated at :-

1.9 x 1019 years

If we assume that there are 64 generations between now and the start of the Christian Era

Then you will have approximately

1.8 x 1019 ancestors

There is a story about an Indian temple which contains a large room with three posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end

if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 264−1 seconds or roughly 585 billion years or 18,446,744,073,709,551,615 turns to finish

ie 1.8 x 1019 moves

The world will certainly have ended !!!

Just a little larger are the 4.3 x 1019

different starting positions for a Rubik's Cube

Any ideas of the smallest number of moves required to solve the cube from any starting position ?

(Known as God's Number)

Any cube can be solved in 22 moves at most, but there is strong evidence that only 20 moves are required.

Hence God's Number is either 20 or 22

Consider this 200 digit number

91474397281474512894803677416201430283564210503433285339561327276933454229609304646471925094518114771016258896592907441426349897556504145570960203925503679105245199142338806082494254050610000000000000

You now have 70 seconds to extract its 13th root

assuming that you wish to claim the world record

The answer is

2,407,899,893,032,210

Compare this to :-

Age of the universe13.798×109 yearsor 4.354×1017 seconds

Number of elementary particles in the observable universe

between 1080 and1097

But these numbers are mere beginners

• Googol

• Googolplex

• Skewe's number

• Gijswijt's sequence

• Ramsey Numbers

• Graham's Number

Googol

Originally named by Edward Kasner

a Googol = 10100

ie

1000000000000000000000000000000000

0000000000000000000000000000000000

0000000000000000000000000000000

(The search engine Google copied the named but

spelled it incorrectly)

Any idea what's special about this number ?

257885161-1

this is approximately 1010000000

It's the largest prime number currently know

It was discovered in 2013 and is some17.5 million digits long

It is also the 48th Mersenne Prime

ie of the type 2n-1

It's the largest known number that cannot beexpressed in terms of smaller numbers

Googolplex

There isn't enough room or time to write out this number

If you can write 2 digits/second, you would need 1.5 x 1092 years, which is some 1082 times greater than the age of the universe

1010

100

or 10googol

Skewes' Number

• Gauss generated a formula to give the number of Primes up to any given number.

• It is known that it generally over-estimates the true number.

• Stanley Skewes showed that Gauss's formula would under-estimate when we got above the number that bears his name

This result, however depends upon the Riemann Hypothesis being true

If the Riemann Hypothesis is ever shown to be false, then Skewesnumber increases to

Which is

1010

1034

1010

10963

Gijswijt's sequence

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3,

2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2,

3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 1, 2, 1, 1, 2, 2,

2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2,

2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2,

2, 3, 3, 2, 2, 2, 3, 2, 1.............

where does 4, 5, etc appear ?

The first time 3 appears is in the ninth

position.

4 appears for the first time in the 221st

position.

The first 5 occurs in position

10100000000000000000000000

ie 1010

23

The other numbers will eventually appear,

although with no sense of urgency

Ramsey Numbers

You are going to have a party.

How many people do you need to invite to be sure that at least 3 people will be mutual aquaintances, or at least 3 people will be mutual strangers ?

However, if we need to know the minimum

number of guests such that either 5 people

are mutually aquainted or mutually strangers,

there are some 2903 or 6.7 x 10271 cases to

try out using brute computer force alone !

We do know that between 43 and 49 guests

would be required

Graham's Number

If we now considerRamsey's numbers butventure into multi-dimensional spacethe numbers get verylarge.So large, that we needa whole new way torepresent them

Once again, say we have some points, butnow they are the corners of an ndimensional hypercube. They are still allconnected by blue and red lines. For any 4points, there are 6 lines connecting them.Can we find 4 points that all lie on one plane,and the 6 lines connecting them are all thesame colour?Graham's Number gives an upper limit of thepoints needed for this to be certain

Knuth's up-arrow notation

Invented by Donald Knuth in 1976It uses a series of up arrows and looks like :-

g64=3↑↑↑↑ g63 times ↑↑↑↑3

If each digit could occupy a Planck length 10-35m, then you could get

1035 x 1035 x 1035 digits ( ie 10105) in a cubic meter, there would still not be enough space in the universe to write it down

We do know the last 500 digits of his

number - the last digit is a 7

Also the lower limit is 13