28 April 2021
Counting possible chess moves,
atoms in the universe, grains of
sand and the number of stars
In the Biblical story of Abraham in the Book of Genesis, Abraham is worried about his survival, his future, and what is going to happen after he dies, for he has no children and so has no heirs.
God brings Abraham outside and says to him, ‘Look towards heaven and count the stars, if you are able to count them.’ Then he said to him, ‘So shall your descendants be’ (Genesis 15: 1-5)
In the Psalms, we are told that God’s counsels are ‘more in number than the sand’ (Psalm 139: 17-18), and if were to count them all we would still be in God’s presence. It is a majestic image of the scope of God’s presence.
But, how many stars are in the sky?
And, how many grains of sand cover the earth’s beaches?
Did you ever look up on a clear, moonless night and ask how many stars can I see above?
When you look up into the night sky it stretches a pitch-black canvas washed with streaks and studs of brightness. We are surrounded by light that has travelled the expanse of the universe to reach our eyes. And it makes me feel tiny and enormous at one and the same time.
But how many stars do I actually see?
There is really no definitive answer to this question. No one has counted all the stars in the night sky, and astronomers use different numbers as theoretical estimates.
Considering all the stars visible in all directions around Earth, some estimates say there are between 5,000 and 10,000 visible stars. But that’s just the stars visible to the naked eye tonight.
But why limit my calculations and my imagination to my own failing, short-sighted pair of eyes?
Why should I simply marvel at the majesty and mystery of it all when I can do some calculations and think of how many stars are visible to God?
Let me start with the galaxies. Astronomers estimate there are around 170 billion galaxies in the observable universe, stretching out over a radius of some 45.7 billion light years.
Those galaxies vary in terms of the numbers of stars they contain. Some galaxies have more than a trillion stars. Some giant elliptical galaxies have 100 trillion stars. There are also tiny dwarf galaxies – tiny, of course, is a relative term here – some tiny dwarf galaxies that have significantly fewer stars.
On the other hand, the Milky Way, our little corner of the observable universe, has 400 billion stars alone.
So, if we multiply the estimated average number of stars in each galaxy by the number of galaxies in the observable universe – and carry the billion, &c – I get a rough estimate of all the stars I am capable of observing. And what I find is there are roughly a septillion stars in the observable universe. That brings us to 1,000,000,000,000,000,000,000,000 stars (1024, or 1 followed by 24 zeros). Which is, well, put simply, an awesome lot of stars.
Other astronomers calculate that there are 10 stars for every grain of sand, 11 times the number of cups of water in all the Earth’s oceans, ten thousand times the number of wheat kernels that have ever been produced on Earth, and 10 billion times the number of cells in a human being.
This is a staggering number: 70 sextillion (or 7 followed by 22 zeros or 70 thousand million million million) stars in the observable universe.
This too is probably a very, very low estimate because the number of galaxies filling the Universe is thought to be much larger than those the Hubble can see.
In his 1980 bestseller, Cosmos, the astronomer Carl Sagan wrote that there are more stars in the heavens than all the grains of sands covering the world’s beaches. He calculated that a handful of sand contains about 10,000 separate grains.
So, how many grains of sand cover the earth’s beaches?
Some years ago, researchers at the University of Hawaii tried to calculate this number by dividing the volume of an average sand grain by the volume of sand covering the Earth’s shorelines.
The volume of sand was obtained by multiplying the length of the world’s beaches by their average width and depth. The number they calculated was seven quintillion five quadrillion (that is 7.5 followed by 17 zeros or 7.5 billion billion) grains of sand.
Mrs Elm, a character in Matt Haig’s novel, The Midnight Library, provides a commentary on a game of chess. She says points out that at the beginning of a game, ‘there are no variations. There is only one way to set up a board.’ There are 9 million variations after the first six moves. After eight moves, there are 288 billion different positions.
‘And those possibilities keep growing,’ she says.
‘There are more possible ways to play a game of chess than the amount of atoms in the observable universe,’ she tells Nora as she lets her win the game.
This value, known as the Shannon Number, represents all of the possible move variations in the game of chess. It is estimated to be between 10111 and 10123. By comparison, there are 1081 atoms that make up the known universe.
The Shannon Number, named after the US mathematician Claude Shannon (1916-2001), is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves.
Considering chess is a human invention, and that it allows us to imagine something greater than the number of atoms in the observable universe, how much more majestic, divine and sublime is it to consider the number of stars and the grains of sand?