For a long time it has boggled my mind how when I sit down to play a game of chess, that particular game will inevitably end up being a completely unique game that has never been played before in human history. After all, it is a fairly defined and limited game: 16 pieces for each side, 64 total squares. And, yet, as we have seen in Parts 1 and 2 of this series, there are far, far, more possible chess games than there are atoms in the universe. In fact, the number of atoms in the universe (1080) is a ridiculously small fraction (10-40) of the Shannon Number (10120), a rough estimate of the total games possible.
It is incredibly complicated and probably impossible to put an exact number on possible games, and anyways, a likely more meaningful concept is the number of reasonable games of chess that are possible. By reasonable, I mean that both sides have a basic competency in chess and are each seeking to win in the best way they know how. By one estimate, this drastically reduces the possible number of games to around 1048.
1048 is a pretty easy number to arrive at: if we assume that for any given turn, a player has an average of, say, four reasonable moves available, and if the typical game lasts 40 moves, that gives 480 = 1.5 x 1048. (40 moves means each player has 40 turns, with 4 reasonable options at each turn; the 40 move average has been corroborated a number of times through the years and was used by Claude Shannon in deriving the Shannon Number.)
1048 is a dramatically smaller number than the Shannon Number, or even the number of atoms in the universe. Somewhat coincidentally, this number is quite close to the fairly well-known number of atoms on planet Earth: 1.4 x 1050. Lest we think 1048 is a much more manageable number than the Shannon Number, however, or any less sublime a result, consider the following: if we were to count all 1048 games, 1 per second, it would take us 2 x 1030 ages of the universe (which is 14.7 billions years). This is also difficult to comprehend, though, so let’s look at the average number of atoms in an average human body: roughly 7 x 1027 atoms. Counting those atoms 1 per second would only take 15 billion ages of the universe, and if we did that another 140 million trillion times, we’d be at a number that equals the total reasonable chess games possible. Or, using an illustration from Part 1 of this series, counting the number of reasonable chess games is equivalent to counting all the molecules of water in Lake Winnipeg…100 millions times!
Of course, my estimate that a decent player is weighing four different moves on average per turn is quite subjective. In fact, in a future post I will argue, based on empirical data, that expert and master level players are deciding between only two different moves on average, giving a much lower number of reasonable games (6 x 1024 , which is still a ridiculously large number: 10 moles of possible games). On the other end of the spectrum, it is clear that weaker players are considering a much larger number of moves, perhaps 10 at each turn. There are on average 30 legal moves per turn, but most players will not consider moves that give their pieces away or have no part of any coherent strategy at all, so 10 seems like a reasonable estimate for safe moves that have some purpose. This gives a total of 1080 total possible reasonable games for non-expert players. Coincidentally, this number is the same as the total atoms in the universe!
I will conclude by briefly considering the same question for another popular and somewhat similar game: Scrabble. A small survey of top-level tournament play in Scrabble indicates that one player scores 424 points on average per game with an average of 28 points per turn, so that gives an average of 15 moves per game. Now, in the same vein of reasoning I used above for reasonable chess games, from purely subjective experience I would estimate that on any given move a player is seriously considering perhaps five different plays. That might represent 3 words, 2 of which can be played two different places, 5 different words each only playable in one spot, etc. 5 options seems reasonable, though it might be 4 or 6 depending on the player.
At first glance, then, it might seem that there are only 530 = 9.3 x 1020 reasonable Scrabble games. However, unlike chess, we have to account for the great variety of ‘starting positions.’ In chess, there is only one fixed starting position (except in “Fischer random chess,” also known as Chess960), but in Scrabble the game begins by each player randomly drawing 7 letter tiles from the bag of 100 tiles. It’s a fairly complex statistical problem to calculate the total number of unique starting racks of 7 tiles for two players. It can be shown that, given all the letter distributions, e.g. there are 12 E’s, 9 A’s and I’s, etc., there are 3.2 million unique sets for the first 7 tiles drawn from the 100.* Calculating the possibilities for the second player’s draw of 7 tiles is much more difficult, however, because it depends on all the possibilities for the first 7 tiles taken and what may or may not be left in the remaining 93 tiles, e.g. if 2 N’s and no E is taken in the first 7, that is much different than if 5 E’s and no N were taken. A good estimate on the total number of possibilities for the two starting racks is 6 trillion (6 x 1012), which means that the total number of reasonable games is 5.4 x 1033.** The range on this number hinges on how many moves a given player could be seriously tempted to play and also the total number of moves in the game. An upper limit for the total number might be for 20 moves and 7 plays contemplated per turn–that would give 740 = 6.4 x 1033 games from a starting set of tiles, and 3.8 x 1046 total possible games.
It would seem chess gives more total possible reasonable games, owing entirely to the greater number of moves in a game, but Scrabble has a sublimely large number of possibilities, as well. It is reassuring, yet strange, that such well-defined, limited games dealing with relatively small numbers (100 tiles, 32 pieces) yield possibilities that rival the number of atoms on our planet, and even in our universe.