The Eight Queens Puzzle 

After discovering all the mathematically possible games in chess, it’s important to look at the mathematical puzzles. We’ll start with the most famous chess problem: The Eight Queens Puzzle. Max Bezzel published this puzzle in 1848, and Franz Nauck published the first solutions in 1850. Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n×n squares. It is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; therefore, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. However, considering that some solutions only differ by rotational and reflectional symmetry, the puzzle only has twelve fundamental solutions.

The Knight's Tour 

Another mathematical chess problem deals with the oddest piece, the knight, which has a unique movement: it moves two squares vertically and one square horizontally. This problem involves the following question: can the knight move through all of the squares on the board while landing on each square only once? This is a very ancient problem, which was first mentioned by the Arab chess player Al Adli ar-Rumi in 840 AD In his book, “كتاب الشطرنج” (“Book of Chess”), where he presents two possible solutions for this problem. A lot of mathematicians have tried solving this problem, only to find that it has considerable solutions; however, none of them could find the exact number of solutions until 1997, when Australian mathematician Brendan McKay managed to find a better answer. McKay found that this problem has no less than 1,658,420,855,433 possible solutions without counting the symmetrical ones.

The Square Rule  

After all these big numbers and complicated chess problems (which are largely theoretical), it’s vital to finalize with something more simple: geometry. The square rule is a visualization tool used to quickly assess if a pawn can promote itself before the enemy king threatens it. As the name implies, it functions by visualizing a square on the board. The pawn can promote if the opposing king can't step into the square on its next move. The first step is to imagine a diagonal line that starts on the pawn and ends on the promotion rank (to the side where the enemy king is). Then, the player must imagine another line starting at the ending square of the previous line and extend it all the way down the file to the rank of the pawn. Then complete the square as shown in the figure. This rule states that the king can capture the pawn if it can step into the square on its next move. If that isn’t the case, then the pawn can reach the promotion square before the king gets to it.