01/03/2023

In Folha: the enigma surrounding the knight's move on the chessboard.

Foto: Leonhard Euler

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

"I found myself, one day, in a group where, during a game of chess, someone proposed the following challenge: to move a knight along the squares of the chessboard, never passing through the same square twice, and starting from a given square." Thus, in a work written in 1759, but published only seven years later, Leonhard Euler recounts how he became aware of "a curious problem that defies any kind of analysis."

Euler is the most prolific mathematician in history: the collection of his complete works began in 1908 and, more than 80 volumes published later, is still not finished. Even so, this work is among his best known, and for good reason: as often happens with the "curious" problems addressed by Euler, this one unfolds into a myriad of questions that delve deep into mathematics, and far beyond.

The problem was not new: the first known mention dates back to the 9th century and has a very curious form. It is the poem " Arrangement in the Steps of a Horse ," written in Sanskrit by the poet Rudrata, from the Kashmir region. It is one of the first Indian texts to make reference to chess. It has the form of a "half-chessboard" composed of four verses of eight syllables each and can be read either sequentially, from left to right, or by skipping syllables according to the movement of the knight in chess.

Forma como pode ser lido o poema ‘Arranjo nos Passos de um Cavalo’, do poeta indiano Rudrata (séc. 9°) – Reprodução Folha

This format has been cultivated by Indian poets for centuries, to such an extent that it is not surprising that one of the most interesting solutions to the problem of the horse's journey is also presented in the form of a poem. It appears at the end of the fifth volume of the "Bhagavantabhaskara," a kind of encyclopedia on rituals, law, and politics written by Bhatta Nilakant'ha sometime in the 17th century, at least 60 years before Euler.

Despite his initial perplexity, Euler approached the problem systematically and discovered an ingenious method for finding different solutions. He found that there are countless possibilities for the horse's journey, "although the number of possibilities is not infinite, it is so great that you will never be able to exhaust it," he stated, which seems somewhat contradictory.

Uma dos trilhões de possibilidades para o ‘desafio do cavalo’ no xadrez – Reprodução Folha

To read the full text, visit the newspaper's website.