.
Today I made a calculation which really surprised me.
It concerns the number of ways a pack of cards can be
shuffled...
The number of ways is equal to:
52!, or approximately 8.07 ×1067 52!, or 52 factorial, means
52 ×51 ×50 ×49 ×¼×3 ×2 ×1
To work this out, I used the factorial button on my calculator and rounded
the answer to 3 significant figures. You could also use Stirling's approximation (see the Notes). The exact figure would certainly have a number of noughts at the end but
certainly not 67 of them.
You might like to work out just how many zeros there actually are at the end
of the number 52!
But just how big is this number?
I wanted to get an idea of what this might be like and decided
that a good way to do that might be to consider how long it
would take to recreate every possible shuffle. I had this
feeling that I might need some help so thought about how many
people I might need to help me get it done in a reasonable time
span.
To make things easy, let us assume that a person would be able
to recreate one new shuffle every second.
There are 60 seconds in a minute, 60 minutes in an hour and 24
hours in a day. So...
In one day one person could generate:
60 ×60 ×24 = 86400 = 8.64 ×104 shuffles.
There are 365 days in a year, so if I worked non stop I could manage:
365 ×86400 = 31536000 = 3.15 ×10 7 shuffles in one year.
Still feels like a long way to go! I need a lot more time so let's think
about
the age of the universe, that might help if I could live that long!
The universe is estimated to be between 12 and 20 billion years old.
Let's
take its age, for the sake of this calculation, to be 15 billion years old.
That is 1.5 ×1010 years.
This means that, if I started doing the shuffles when the
universe began, I would still be doing them now and I would
have done:
3.15 ×107 ×1.5 ×1010 = 4.73 ×1017
shuffles.
I think I will need some help!
Our galaxy contains approximately 100,000,000,000 = 1.00 ×1011 stars (suns). There are approximately 100,000,000,000 = 1.00 × 1011 galaxies in the universe.
So there are approximately 1.00 ×1022 stars in our universe. This is looking promising! But I think we still have some way to go.
Let's assume that around each star there circles one earth-like
planet (that is a planet with alien life on it). This would mean
that the number of Earths in the universe is
1022.
There are approximately 7 billion people living on the Earth
(7.00 ×109). So there could be 7.00 ×109 ×1.00 ×1022 = 7.00 ×1031 aliens living in the
universe.
So, I could utilise the population of the universe to help me!
Let's imagine that all these aliens have been living since the
big bang and have been shuffling away. All they have been doing
since the beginning of time is shuffling their pack of cards -
one permutation per second.
The number of permutations tried by these aliens is therefore:
7.00 ×1031 ×4.73 ×1017 = 3.31 ×1049
Which is only a fraction of all the possible permutations of a
single pack of cards.
This is possibly why we have not been contacted by aliens yet -
they are all too busy shuffling packs of cards!
