Math Tricks found in Chess

There’s lots of clever math tricks hidden in chess. Here are some examples.

Pawn race: Want to know if you can catch your opponents pawn before it becomes a queen? Use simple geometry! Draw a diagonal line starting from the pawn to the first row, then make a box. If your king can enter that box, he can catch the pawn! No need to count moves or calculate.

Opposition: Want to know if you have the distant opposition in a king and pawn engame? Well you could calculate it out, or you could use simple parity! If the king’s are on the same line (diagonally, horizontally, or vertically) look at how many squares are in-between the kings. If it’s your turn: an even number of squares means you have the opposition, an odd number of squares means your opponent has the opposition. There’s six squares between the kings in the diagram below. If it’s white’s turn, they have the opposition!

Calculating the relative value of pieces:

If you randomly place a piece on an empty chess board, how many moves will it have on average? I calculated this out a long time ago, then I compared it to what the pieces are assumed to be worth.

This is over-simplistic, but interesting. So it looks like the king has a relative value of 4, and Bishop’s are slightly better than knights. The downside of only looking at available moves is that it ignores things like the fact that the Bishop can only attack one color square, or how pieces work together. The Bishop pair, for example, is worth more than the sum of each individual bishop, as they complement each other. This fact is used in chess engine programming.

Shortcut to calculating knight moves:

I use this trick all the time in my games. Want to know how long it takes a knight to get somewhere? Simply memorize this square. The knight square is a beautifully symmetrical pattern, which makes it really easy to memorize. Each number represents the number of moves it will take the knight to reach that square.

Interesting fact – If you wanna reach any of the fours, then any move the knight makes will get you closer to reaching that square.

Knight’s also operate according to parity. That is, they attack dark squares, then light squares, then dark squares, and so on. See the pattern above – even moves get you to the same color square that your sitting on, odd moves get you to the opposite colored square.

Pythagorean Theorem does not apply in chess:

Our brains are hardwired to think that diagonals are longer than edges of a square. This is not the case in chess. This can lead to some interesting positions. For example:

White to move. It looks like black is going to promote their pawn and white will lose. If white tries to promote their pawn right away the black king will catch it. Miraculously, white can actually force a draw here. They do so by marching their king diagonally as shown. They will either catch the black pawn or promote their white pawn. This is only possible because it takes the same number of moves for white to move their king from h8 – e5 t- h2 as it does to go in a straight line from h8 to h2. Counter intuitively, going diagonally doesn’t increase the trip length.

Centralization matters! Except for rooks…

(Almost) Every piece in chess will have more available moves in the center of the board than they will on the edge of the board. More available moves = more control. This is why chess strategy tells you to always control the center.

This doesn’t apply to rooks… Like the honey badger, rooks just don’t care. If you place a rook on an empty board, they will always have 14 moves. Doesn’t matter where you place them. This is due to the square symmetry of the board, which lines up with the way rooks move.

The 8 Queens Puzzle:

How many Queens can you place on a chessboard such that none of them could capture each other? 8 of course. After all, a chess board is just an 8 by 8 grid. But trying to find the solution is much more difficult. There are actually 96 solutions (12 if you don’t count symmetry or rotations). This is a complex math and computer science problem. They have solved the same puzzle for every solution on a 27 by 27 sized board.

Counting the number of possible games in chess:

A clever mathematician named Claude Shannon calculated the upper bound limit for chess, resulting in about 10^120 possible different games. This was a huge influence on the field of computer science and chess, and shows why we cannot solve chess using brute force.

Shannon number – Wikipedia

There are many other interesting math tricks hidden in chess. I look forward to reading about others.