Chess isn't really a game of strategy once you look at the numbers
Most people play chess and think about it as a battle of wits, positional understanding, and intuition. Those things exist, sure. But the foundation underneath all of that is just discrete mathematics, combinatorics, and probability theory dressed up in wooden pieces. The Coolmaths website covers this angle because it's arguably the more honest way to look at the game. At its core, chess is a finite state machine. Every position has a fixed set of possible moves. From the starting position, White has exactly 20 legal moves. After one move by each side, you're looking at roughly 400 possible positions. By move four, that number balloons to over 288 billion. That's a Shannon number, named after Claude Shannon, who calculated in 1950 that the game tree complexity of chess sits somewhere around 10^120. For context, there are fewer atoms in the observable universe than that. You can't brute-force your way through it. What that means in practice is that chess engines don't search the entire game tree. They use alpha-beta pruning, which is basically a smart way of cutting branches off the decision tree before you even fully evaluate them. If a move leads to a position that's clearly worse than one you've already seen, you stop calculating it. This is why modern engines like Stockfish can evaluate positions at a rate of tens of millions of nodes per second on consumer hardware without actually solving the game.
The evaluation function itself is where things get interesting. An engine doesn't "know" chess. It assigns a numerical value to any given board state based on material balance, piece activity, king safety, pawn structure, and dozens of other features. The weights on those features come from training on millions of games, mostly through self-play. This is the neural network component that changed everything when AlphaZero showed up in 2017. Before that, engines were predominantly tree-search based with handcrafted evaluation functions. After that, the best engines in the world were using a hybrid of neural nets and traditional search. I spent several months working on a small chess analysis tool a while back, and the edge case that almost broke me was handling endgame tablebases. When you get down to three or four pieces, the endgame is already solved. There are precomputed tables that tell you the exact optimal move from any position. The problem is storage. A full 7-piece tablebase is somewhere in the neighborhood of 300 gigabytes. I tried loading it into memory on a machine with 64GB and it refused to cooperate. The workaround was switching to on-demand tablebase lookups where the engine would fetch only the specific positions it needed rather than loading the whole thing at once. That cut the memory footprint from impossible to roughly 8GB. It's not elegant but it works. Here's something most beginners miss about the math side: positional chess concepts like the bishop pair or pawn structure aren't just poetic descriptions. They're measurable features that feed directly into the evaluation function. A bishop pair might add 0.5 pawns of value in an open position and 0.1 pawns in a closed one. The difference matters at engine level because it's the margin between a positive and negative evaluation that determines whether a move gets pruned from the search tree. So when a human player says "I prefer my two bishops," they're essentially expressing a heuristic that an engine would quantify as a weighted feature in its evaluation function.
Another thing people don't think about is the probabilistic nature of chess at lower levels. At the master level, games are closer to deterministic because both sides are playing near-optimal moves. But at the club level, you're dealing with a distribution of possible moves for each position, not a single best line. This is why puzzles and tactical training matter so much more than opening theory for improving players. A tactic is just a forced sequence that exploits a local violation of the evaluation function. Your opponent's brain makes a calculation error, the engine evaluation shifts by several pawns in one move, and the game is over. Coolmaths breaks this down with some good interactive examples, particularly around the combinatorial explosion of possible games and the probability of certain patterns occurring. The fork, pin, and discovered attack are all just local optimization problems in game tree search terms. A fork is a single move that creates two threats simultaneously, which means the opponent has to resolve both or lose material. In search terms, it's a node where the branching factor effectively doubles for the defender. The practical takeaway is that if you want to understand chess better through math, start with calculating tactical motifs. Learn to spot forks, pins, and skewers quickly. These are the areas where the mathematics is most concrete and immediate. Positional play is harder to quantify and harder to improve through analytical study alone. It comes from pattern recognition built over hundreds or thousands of games.
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There's a limit to how far the math can take you though. Even Stockfish, the strongest engine in the world, doesn't "understand" chess. It searches. It evaluates. It picks the move with the highest expected value according to its evaluation function. It has no concept of long-term strategy the way a human grandmaster does. The illusion of understanding comes from the fact that the evaluation function captures enough features of the position that the moves it selects look strategic to a human observer. That's a useful distinction to keep in mind. If you want to explore this yourself, the Coolmaths page on chess mathematics is a solid starting point. It walks through the basic combinatorics and probability concepts without assuming much background. For something more technical, look into how MCTS (Monte Carlo Tree Search) was used in early versions of Leela Chess Zero. It's a completely different approach from Stockfish's alpha-beta search and it was surprising how well it performed despite being conceptually simpler. The tradeoff is that MCTS requires more computation time per move, which is why Stockfish still holds the rating title on traditional hardware. The math won't make you a better player overnight. But it will change how you think about the positions you see on the board. Instead of asking "what's the best move?" you start thinking in terms of branches, evaluations, and pruning. It's a different lens. Not necessarily better, but useful.
