The compelling entwinement between Chess and Math

Needless to say, during the Covid-19 pandemic and the numerous lockdowns we have faced consequently, it has been very hard to not be bored while stuck at home all day. As a result, along with the booming success of Netflix’s original series: The Queen’s Gambit, in October 2020, chess has suddenly become extremely popular. 

Generations of people have been fascinated in the mathematics deeply rooted in the game. For example, something as simple as deciding whether to make a trade or not involves doing maths. By working out the relative value of your pieces compared to your opponent’s, you can make an informed decision after some basic arithmetic.
The mathematics of chess and the century-old theories and problems that incited legendary mathematicians, such as Euler and Gauss, still gain interest today. This essay will take you through some of the most popular  mathematical problems that have arisen from the historic game of chess. 

Do you know that the no. of possible chess games at move 40 exceeds the no. of atoms in space? Well, That is unsurprisingly true. The number of atoms in the universe is 10^80(1 with 80 zeroes next to it) while the number of chess games is 10^120 according to the American Mathematician Claude Shannon. To illustrate this, At move 5, There are 69,352,859,712,417 different positions.

So, one could ask, What type of computers do the world's top 10 players use to prepare for games? The answer is a "normal computer completely dedicated to chess". According to Claude Channon,

"With chess it is possible, in principle, to play a perfect game (solving chess) or construct a machine to do so as follows: One considers in a given position all possible moves, then all moves for the opponent, etc., to the end of the game (in each variation). The end must occur, by the rules of the games after a finite number of moves (remembering the 50 move drawing rule). Each of these variations ends in win, loss or draw. By working backward from the end one can determine whether there is a forced win, the position is a draw or is lost."

There is only one way to solve the game. Also known as brute force ( to check all the possibilities). Imagine checking possibility of the 10^120 positions to be perfect. This process is tiring in terms of available resources.

That is why Endgame Tablebases were brought to reality. Instead of checking all possibilities with all the pieces on the board, Only 2-7 pieces are left. Tablebases have solved chess to a limited degree, determining perfect play in a number of endgames, including all non-trivial endgames with no more than seven pieces or pawns (including the two kings). An endgame tablebase is a computerized database that contains precalculated exhaustive analysis of chess endgame positions. Interestingly, The longest seven-piece example is a mate-in-549 position discovered in the Lomonosov tablebase by Guy Haworth

Can you find the mate in 546 for Black?

Answer: This position is beyond human and computer capabilities, even with access to a database. It would take 140 TB of storage space to begin the solution.

The way tablebases work is through retrograde analysis. Retrograde analysis is a technique employed to determine which moves were played leading up to a given position.

Do you even imagine that the intuitive move g4 is a blunder? This will allow the black king to oppose the pawn and shoulder the white king off. That is why endgames is the core of chess. 

The advantage of tablebases is that istead of analyzing forward from the position currently on the board, the database would analyze backward from positions where one player was checkmated or stalemated. Thus, a chess computer would no longer need to analyze endgame positions during the game because they were solved beforehand. It would no longer make mistakes because the tablebase always played the best possible move.

2- Elo Rating

Your Elo rating will fluctuate depending on the results of your games. The 
lowest possible rating is 100 (the mathematics would allow you to go lower but 
the International Chess Federation (FIDE) has restricted the value to 100). The 
highest rating ever reached in classical chess is current world chess champion,
GM Magnus Carlsen’s rating of 2882 in 2014.

So, what do these numbers actually mean? The difference in the ratings between two players serves as a predictor of the outcome of a match. Two players with equal ratings who play against each other are expected to score an equal number of wins. A player whose rating is 100 points greater than their opponent's is expected to score 64%; if the difference is 200 points, then the expected score for the stronger player is 76% and so on. If Player A has a rating of RA and Player B has a rating of RB, the exact formula (using the logistic curve) for the expected score (EA), from 0 to 1, of Player A is: 

That is all for today, Hope you enjoyed the aricle so far. I have avoided some intricacies since it it a complex subject and still under research today.