Re: are there more moves on a chessboard then atoms in the universe

Question: are there more moves on a chessboard then atoms in the universe
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No one really knows the exact number of moves on a chessboard or the exact number of atoms in the universe. Historically the number of moves on a chessboard has been overestimated, while the number of atoms in the universe has varied wildly. Given the state of such estimates today, it looks like there are vastly more chess games out there than atoms in the universe. But as little as a century ago, the answer would have been the opposite. Here my understanding of “moves on a chessboard” as the total number of possible games that can arise when playing the game of chess. These are sorts of “How Many?” questions are ones that people have been trying to answer for quite a while.

Let's start with chess. Believed to have been invented in India in the 6^th century, it has spread worldwide because of the inherent simplicity of the game. With only 64 squares on the board and 32 pieces of 6 different types, chess is a perfect example of what is known in mathematics as the combinatorial explosion effect. With this small number of pieces, Hardy estimated the number of possible chess games at 10 raised to the power of 10 raised to the power of 50. That is a 100 followed by a billion billion billion billion billion billion zeros. Such a number is difficult to grasp, since it is so far outside the realm of human experience. Even after just 40 moves in the game of chess, Shannon estimated there are 10 raised to the 43 power (10 million billion billion billion billion) possibilities. Mathworld has an excellent article on chess with references to the above results.

Since there are so many possible games, it is unlikely that an exact method of counting the number of moves on a chessboard will be found anytime soon. At some point in the future advances in mathematics might allow such a computation, but for now the exact value is completely unknown.

Now on to the number of atoms in the universe. An atom, as you are most likely aware, is a nucleus made up of protons and neutrons surrounded by an electron. Protons, neutrons, and electrons are examples of elementary particles, so for cosmologists, the question is usually phrased as “How many particles are there in the universe?” Archimedes asked a related question in his work, The Sand Reckoner. Believing that a grain of sand was the smallest particle, Archimedes wondered how many such particles would fit in the universe. Of course, the universe for Archimedes only went out to the fixed sphere that the stars were believed to rotate on. In The Sand Reckoner, Archimedes found that 10 raised to the 63 power number of grains of sand could be packed in the physical universe.

From the time of Newton (in the 17^th century) up through Einstein, a different model of the universe was considered. In this universe the stars just went on and on and on forever. On the other hand, physicists such as Einstein knew that this was just a model, that it might not be true. Einstein perhaps said it best when he joked, “Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.” So Einstein had his doubts about the infinite universe theory. But under this model (and remember, it lasted around three hundred years) there is no number of particles in the universe, there are an infinite number. Even the chess board can't match that.

But more recent theories of the universe have a different idea. In these models, the gravity of the mass inside the universe warps space in such a way that the universe is closed around itself. For instance, you can walk forever along the Earth's equator in one direction since after a while you will get back to where you started eventually, even though the Earth is of finite size. In the same way, it might be possible to travel in one direction in the universe and return to where you started. Such a curved universe might have a finite number of particles.

Now, whether or not the universe is curved depends and whether or not there is enough mass in the universe to bend space in upon itself. So the number of particles in the universe (relative to its size) is a fundamental question in cosmology right now. If the universe is finite, the chess board probably wins, since packing the visible universe with neutrons (the modern version of The Sand Reckoner) only yields 10 to 128^th power particles, but the number of chess moves is more than 10 raised to the billionth power five times in a row; chess moves definitely wins in this scenario. The only thing we are sure of about the universe is that the more we discover about the vast reaches of space, the more suprisinging the universe turns out to be.

Mark Huber