Considering its formalization in medieval Europe, chess has some unexpected quirks. The pieces are named for positions in the royal court, but the most powerful piece – and the most valuable – is not the king, but the queen. A knight and a bishop have equal value, just as in the UK House of Lords; a rook, symbolizing the heavy armored divisions of the ancient cultures who originally invented the game, outguns both.
The king, in contrast, is basically the Princess Peach of the game. He’s important, sure – but only insofar as his capture will end the game. Outside of that role, however, he’s pretty useless: weak, limited, and slow.
It sucks for his highness, but at least he had an air of mystery to hide behind – until now. In a preprint paper that does not yet appear to have been peer-reviewed, Christian Táfula Santos, a doctoral student in the University of Montreal’s Department of Mathematics, has formalized just how impotent the king is by measuring his speed against one of his knights. The result: he’s barely more than half as fast as the pony patrol.
Specifically, the ratio is 24 to 13: if it takes a knight 13 moves to reach a certain square on the chessboard, it will take the king around 24 moves to get to the same square.
It’s fast, but perhaps not as fast as you were expecting, right? But it makes sense: sure, a knight moves three places for every one the king clears, but it does have to move in a prescribed pattern – the king can step in any direction. That comes in handy on certain diagonal paths, where the king can speed up to about two-thirds the knight’s pace – still slow, but enough to bring up the average from a flat half.
It’s interesting for sure, but it wouldn’t be a math project if it didn’t get extended to an absurd degree at some point – and for Táfula Santos, that point comes with the introduction of his so-called “super-knights”.
“The shift from traditional knight to super-knight is based on mathematical generalization,” he explained in a statement. “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.”
The result, unsurprisingly, is a higher ratio between the speeds of knight and king – but it’s more nuanced than that. The relative velocities increase in a very predictable way, following a formula depending on a and b.
Choose the right a and b, and things get even prettier. “Imagine a ‘Fiboknight’,” Táfula Santos said – a knight for which a and b are consecutive Fibonacci numbers. Then, each successive knightly velocity is linked to its predecessor “by the golden ratio,” he explained, “reflecting the behavior of the Fibonacci sequence.”
While the relative speeds of various chess pieces may seem to be a niche question, this link to the Fibonacci sequence hints at just how expansive the research truly is. “My research project extends beyond the chessboard,” said Táfula Santos.
“It makes connections between different branches of mathematics, including number theory, geometry and combinatorics,” he explained, “and it opens up prospects for the study of other objects and movements in spaces with more than two dimensions.”
The paper is posted on ArXiv and is also published in The Fibonacci Quarterly.
