When we play chess, we know that the knight is faster than the king to reach a given square, but what is this speed ratio?
This is what Christian Táfula Santos, a doctoral student in the Department of Mathematics and Statistics at the University of Montreal, calculated, whose demonstration was the subject of a publication on the open science site arXiv, which archives nearly of 2.4 million scientific articles in different fields.
The ratio he discovered is 24/13, that is, the knight is, on average, 1.85 times faster than the king in reaching a square on a chessboard – meaning that, if the knight takes around 13 moves to reach a certain square, it will take around 24 for the king to reach it in turn.
But it is not so much the answer as the reasoning behind this ratio that is special: Christian Táfula Santos draws on the work of mathematician Askold Khovanskii, who describes how certain sets of numbers grow when added, to explore a entire family of “modified riders” and create an unexpected bridge with the famous Fibonacci sequence.
“Superhorsemen” on an infinite chessboard
Christian Táfula Santos
Credit: Jeremy Schlitt
In this demonstration, Christian Táfula Santos replaces the traditional rider and its L-shaped movement with a “superrider” capable of moving a number a of boxes in one direction and a number b of boxes in the other. The “superrider” refers to a case where a et b are coprime and whose sum is odd.
“The progression from the traditional rider to the superrider is part of a mathematical approach to generalization,” he explains. So I broadened the concept by trying to find out what would happen if the rider could move a boxes in one direction and b squares in the other instead of the usual movement.”
In this way, the superrider manages to perform a wider movement, moving two squares in one direction and three in the other, where a = 2 et b = 3. The ratio of this move to the king therefore becomes 90/31, which means that it is approximately 2.9 times faster on average than the king.
“Therefore, it is also mathematically logical to move from the generality to particular cases, by imagining a “fibocavalier”: if a et b are Fibonacci numbers, the resulting speeds are related by the golden ratio – i.e. 1.618… –, reflecting the behavior of the Fibonacci sequence!” adds the doctoral student.
Christian Táfula Santos’ demonstration thus contradicts the intuition that the average speed of the knight is double, since he can reach certain squares twice as fast as the king.
However, on certain diagonal paths, the king is slower: the knight’s speed factor increases to 3/2, or 1.5 times faster on average.
“That said, my research project goes beyond the framework of chess,” concludes Christian Táfula Santos. It establishes links between different branches of mathematics, including number theory, geometry and combinatorics, and it opens perspectives for the study of other parts and movements in spaces with more than two dimensions.
This shows that an age-old game like chess can still reveal unsuspected mathematical properties!
Senegal
