Knight's tour on a chessboard from ancient Sanskrit text

 

My friend Dilip Ranade forwarded the above video along with the following write up:

The video shows a solution to the problem of a knight moving to all squares of a chess board without repeating a square. 

Amazingly, solutions for the same have been found in Sanskrit classics!

Here is a link to a short paper available in an IIT site:

 https://www.iiitb.ac.in/Ganaka/projects/paduka/paduka.html

Here is an elaboration of the theme of the knight's move to cover all squares in a 4x8 chessboard exactly once as derived from ancient Sanskrit Shlokas. Research tells us that there is a Chaturanga-Turanga Bandha corresponding to Śhlokas 19–20, found on page 392 of the Paduka-Sahasram edition.

This is the famous knight’s-tour shloka.

What “Chaturanga-Turanga Bandha” Means:

Chaturanga = chess (the four-limbed ancient Indian game)

Turanga = a horse

So, Turanga-bandha = “the horse-pattern” → in chess this means the knight.

Thus this bandha (poetic figure) is a verse arranged exactly like a knight’s tour on a modified board.

Why This Shloka Is the Knight’s Tour:

In this bandha:

The poet writes a Sanskrit verse.

He then splits the verse into syllables.

Each syllable is placed in one square of a grid.

The squares are numbered.

The numbers show the order in which a knight can visit all squares without repeating any. This is the Hamiltonian path problem on a chessboard. Because the pattern involves a chess-knight (turanga), the shloka is named 'Chaturanga-turanga-bandhaḥ — the chess-knight pattern.'

Here is the Shloka 19:

And, this is how the arranged with syllables in row and column grid by the poet:

The syllables in English letters and the numbers in Roman numerals would be the following table:

How the knight’s movement is used:

As we can see, the squares of the board are arranged in a 4 × 8 rectangle. And each syllable is placed in order row wise. If you read left to right, row wise, you get the Shloka (written in Devnagri)given above. The above matrix gives the sounds of the syllables in English and the numbers in Roman numerals.

If you follow the numbers in the diagram:

1 → 2 → 3 → … is a legal knight move each time. You will cover all 32 squares exactly once. The syllables of the shloka are placed in an order that corresponds to a knight hopping from one square to the next. The syllables of the shloka are placed in an order that corresponds to a knight hopping from one square to the next.

Reading the syllables in that order reconstructs the full shloka. And below is Shloka 20 that has been created by knight's legit moves across the board, and touching each square only once:

And here is the move of the knight to create the above Shloka number 20:

This is a constrained combinatorial construction. The poet must choose a verse whose syllables can be arranged so that the knight's path is possible.

Why this s combinatorially really amazing:

Knight tour of duty are nontrivial. Finding a knight’s tour (even on a 4×8 board) is a known combinatorial puzzle. But here, the poet must also place syllables. He must arrange metrical structure, meaning, and the knight's Hamiltonian constraint simultaneously.

Are there more than one way that the knight could move?

Yes — mathematically, there are many possible knight tours on a 4×8 board. But only one tour matches the syllable order that reconstructs the shloka.