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!
