How much faster is a knight than a king? A mathematical study reveals the exact ratio!

We know a knight can move across the chessboard faster than the king, but exactly how much faster?

Christian Táfula Santos, a doctoral student in UdeM’s Department of Mathematics, has done the math and his proof has been published on the open science site arXiv, which archives nearly 2.4 million scientific articles in various fields.

Táfula Santos concluded that the knight can reach its destination an average of 24/13, or 1.85, as fast as the king. In other words, if it takes a knight about 13 moves to reach a given square, it will take the king about 24 moves to reach the same square.

But the interesting thing isn’t so much the solution as the method. Táfula Santos drew on the work of mathematician Askold Khovanskii, who has described how certain sets of numbers grow when you keep adding numbers from the set together, to create a new breed of chess knight and, surprisingly, link it to the well-known Fibonacci sequence.

Super-knights on an infinite chessboard

Táfula Santos replaces the traditional knight and its L-shaped trot with a “super-knight” that moves a squares in one direction and b squares in the other, where a and b are coprime numbers and their sum is odd.

“The shift from traditional knight to super-knight is based on mathematical generalization,” Táfula Santos explained. “I extended the concept to see what would happen if the knight could move a squares in one direction and b squares in another, instead of the usual pattern.”

This gives the super knight a wider orbit. For example, if a = 2 and b = 3, the knight can move two squares in one direction and three in the other. In this case, the ratio of the knight’s speed to the king’s is 90/31, so the knight is about 2.9 times faster than the king, on average.

“From there, it also makes mathematical sense to move from the general to the particular and imagine a ‘Fiboknight’: if a and b are Fibonacci numbers, the resulting speeds are linked by the golden ratio—approximately 1.618—reflecting the behavior of the Fibonacci sequence,” Táfula Santos continued.

Táfula Santos’ demonstration thus corrects the intuition that, although the knight can reach some squares twice as fast as the king, its average speed is not double. But on some diagonal paths, the king can almost keep up, the knight being only 1.5 times faster, on average.

“However, my research project extends beyond the chessboard,” said Táfula Santos. “It makes connections between different branches of mathematics, including number theory, geometry and combinatorics, and it opens up prospects for the study of other objects and movements in spaces with more than two dimensions.”

Chess may be 1,500 years old, but its mathematical implications remain to be explored!