The mathematical challenge of the knight’s ride in chess – 02/28/2023 – Marcelo Viana

“One day I found myself in a group where, during a game of chess, someone proposed the following challenge: travel with a knight through the squares of the chessboard, never passing through it twice, and starting on a given square “. Thus, in a work written in 1759, but published only seven years later, Leonhard Euler tells 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, it is still not finished. Nevertheless, this work is among his best known, and for good reason: as is often the case with the “curious” problems tackled by Euler, this one unfolds into a myriad of questions that go 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 shape. This 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 board” composed of four verses of eight syllables each and can be read either sequentially, from left to right, or skipping syllables according to the movement of the knight in chess.

This format has been cultivated by Indian poets for centuries, to the point that it is not surprising that one of the most interesting solutions to the horse-riding problem is also presented in the form of a poem. It appears at the end of the fifth volume of “Blagavantabhaskara”, 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 bewilderment, Euler systematically tackled the problem and discovered an ingenious method for finding different solutions. He found that there are many, many possibilities for horseback riding, “although the number of possibilities is not infinite, it is so great that you will never be able to exhaust it”, he said, so that seems a bit contradictory.

Calculations performed at the end of the 20th century using several computers confirmed his intuition. The number of closed horse rides, that is, those that return to the starting point, is estimated at 13,267,364,410,532 (13 trillion…). It is more than the diameter of the Solar System measured in kilometers.

For non-enclosed tours the current estimate is even more colossal: 19,591,828,170,979,904 (19 quadrillion…).

Next week there will be more stories about the horse ride.

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