The greatest number you can imagine

by Prof. Christian Hesse
6/17/2021 – How many different games of chess are possible? Everyone knows it's a very, very large number. It can be written down in seconds, using just a few digits, is unimaginably huge. Unimaginable? The great scientist Enrico Fermi recommended that we at least try to understand very large numbers, to estimate what they entail. Mathematics professor Christian Hesse attempts to do this for the number of chess games. You will be stunned!

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Enrico Fermi was one of the most renowned physicists of the 20th century. He worked on the development of the atomic bomb in Los Alamos during World War II. In Fermi’s opinion every educated person should be able to carry out plausible calculations in order to quickly estimate any size. He asked his students questions like: how many grains of sand does the Sahara have? How much does Mount Everest weigh. In this section we will deal with a Fermi question: how many chess games are there?

As you have probably heard, it is a very large number. But how large? Is it at all imaginable?

But before we come to it, there is another well-known very large number in chess: when the inventor, Sissa ibn Dahir, presented the wonderful new game to his king, the grateful monarch wanted to know how he wished to be rewarded for his invention. 

In false modesty Sissa asked for a single grain of wheat on the first square of the chessboard, two on the second, four on the third, etc., always doubling the number on the following square, until the chessboard was covered. The king agreed, until he realized that 1 + 2 + 4 + 8 + ... resulted in him owing Sissa 264 – 1, which is 18,446,744,073,709,551,615. That is more than eighteen quintillion grains of wheat, about 2,000 times the annual world production. Legend has it that the emperor did not grant Sissa’s request, but instead took his head.

Now back to the question: how many different chess games are theoretically possible?

Games with more than 60 moves are rare, and on average there are 30 legal moves in each position that is on the board. This means there are 30120 possible games. Naturally the vast majority of these games is blatantly nonsensical. But the fact remains that it is 10180 different games.

Now this is a number that has absolutely no relevance to anything in our real existence. The number of elementary particles in the hundreds of trillions of stars in the known universe is unimaginably smaller.

So let us try to modify the number, make it more manageable. To do this we try to define what we will call “meaningful” chess games. We assume that there are only five reasonable moves in an average chess position. In many there may of course be more, and in many less. The average of five moves per position seems reasonable.

Again we will assume that the overwhelming majority of reasonable chess games will be no more than 60 moves long, i.e. a maximum of 120 half-moves. With these two assumptions we arrive at a number of meaningful chess games that is much smaller: 5 x 5 x 5… x 5 with 120 factors. When multiplied, we get a result that is 1080 meaningful chess games. In mathematical terms this is a hundred tredecillion chess games.

But what does it mean?

1080 instead of 10180. What a relief! That, surely, is a more manageable number.

 1080 happens to roughly correspond to the number of atoms in the known universe, which consists of an estimated two trillion galaxies, with a few hundred billion stars in each galaxy.

But is 1080 really a number we can comprehend (in the Fermi sense)? Let us try.

Warning: the passages you are about to read can cause dizziness and severe mental fatigue – so a little caution is advised. Maybe you should put some pillows on the floor in case you faint.

We are going to try to break down the number of reasonable chess games into something we Earthlings can just about grasp – like the number of grains of sand in the Sahara, the weight of Mount Everest, the distance to the moon.

We start by setting up a computer that plays through a million (reasonable) games per second, and task it with do this for a hundred tredecillion (1080) games. After starting it we we walk across to the Sahara. There we pick up one grain of sand. This we transport to Arizona, crossing the Atlantic in a row-boat, and toss the grain of sand into the Grand Canyon.

Mind you, we are very slow walkers, and require quite some time for each step we take. Similarly, we are very slow rowers, and each oar stroke takes a great deal of time. Transporting this single grain of sand takes a hundred years.

As soon as the grain is in the Grand Canyon, we begin a second task. We walk over to Mount Everest and carve out a teaspoon-full of it. This we transport, at the same speed (a teaspoon per century), to Canada. There we deposit it on the ground. Then we return to Everest for a second teaspoon full. And we continue doing this until the entire mountain is standing in Canada. After this we reverse the process, moving Everest, teaspoon by teaspoon, back to Nepal. Once this is completed, we return to our computer. Has it played through the 1080 games, running at a million games per second? Nowhere close!

Now, we walk and row back, at the same breath-taking speed to Africa. We pick up a second grain of sand in the Sahara and transport it to the Grand Canyon.  After this, we go back to Mount Everest and transport the entire mountain, teaspoon by teaspoon, to Canada, and then back to Nepal. Only then we are ready for the third grain of sand. We keep doing this, one grain of sand at a time, and after each grain we move the entire Mount Everest back and forth, until the Grand Canyon is filled with Saharan sand.

Now we proceed to do the reverse: we return all the sand, grain by grain, back to Africa. And between two grains we transport Mount Everest with a teaspoon to Canada and back. We do this until the Grand Canyon is empty. I grant you it takes a very, very, very long time.

In terms of the Sahara and Everest, we are back to square zero. To make a note of this, we take a sheet of paper, one square meter in size, and use a pencil to make a dot in the top left-hand corner. Then we proceed to repeat the entire cycle: filling the Grand Canyon, grain by grain, emptying it again, and between two grains we move all of Everest, teaspoon by teaspoon, and return it to its original place. When the cycle is complete we make a second dot on the paper, next to the first. After this we repeat the procedure: a first grain of sand, all of Mount Everest back and forth, a second grain of sand, etc. We keep doing this until the entire paper is filled with dots. Has the computer now at last finished playing through at least a few tredecillion games? Not yet!

So we repeat the entire process with a second sheet of paper, and when that is filled, with a third, and a fourth, putting one sheet on top of the other. We keep doing this until we have a pile of paper sheets that reaches the moon.

Exponential Growth, from TED talk by Adrian Paenza (animation by TED-Ed)

Then we start erasing the dots, one at a time, filling and emptying the Grand Canyon, dismantling and reconstructing Everest after each grain of sand, before erasing each dot. Slowly we are getting tired.

So please tell us, you may say, when all the sheets of paper are empty, did computer at last play through all 1080 games. No, but we are closing in on the number. We have to repeat the process, with a new pile of sheets reaching all the way up to the moon, filled with dots and then erased – and do this again and again. How often? Ten times or a thousand times? No, in fact we have to make 200.000 similar piles to the moon before we are done. Only then the computer has at last finished its task: playing through all the reasonable games of chess that are possible.

What is the purpose of this deeply outlandish story? It is to give you a faint impression of what really big numbers mean. Saying there can be 1080 possible games of chess is probably a bigger deal than you could ever have imagined.

 


Christian Hesse holds a Ph.D. from Harvard University and was on the faculty of the University of California at Berkeley until 1991. Currently, he is Professor of Mathematics at the University of Stuttgart.

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