The knight rides his math assignment

In the ancient ashtapada form of the Indian chess game, Professor Rangachar Vasantha of Sri Krishnadevaraya University, Anantapur sees a manifestation of cosmological order determined by the play of numbers and spaces.

 It is not just the ordered motions of planets and stars that one can see outside of oneself in the sky.  It is rather an internal process which results in the superimposition of one’s own calculations on the movements of entities even though conforming to their sets of rules. Predictive sciences of those days such as astrology that depended on the movements of planets could have inspired the design of the game too, the learned Professor seems to imply.


It is just possible that the dependent attitude of fatalistic predetermination, which is the essence of a client of an astrologer, gets a break, when the man prevails as the master of the board rather than becoming either the subject or the handmaiden of a chart. The same numerical skills that go into the casting and analysis of horoscopes serve also the player in chess, while however raising him from the position of a victim or an interpreter to that of a manipulator.


And you get to manipulate one royal army against another, no less! Forgetting for the moment the modern nomenclature of the chessmen, let us remember that ratha (chariot), gaja (elephant), turaga (horse) and padaati (foot soldier) forces were moved on the board resembling the formations and movements of the armies of the kings of those times. Rules of their movements were created keeping in mind the might, stature and intrinsic tendencies of these forces. Thus the foot-soldier was always expected to move only forward, in a self sacrificing mode, and to cut either to his right or left forward, namely diagonally. The minister/ army commander could be taken in any direction, several steps at a time. The king moved very little either in aggression or in flight. The chariots oriented themselves and moved non-diagonally while the ponderous elephants moved only diagonally.


The horse rider (knight in modern parlance) was tricky. He seemed to move sideways up to the length of his horse, while actually moving forward or backward one step. Again he may move forwards or backwards one horse-length while moving one step sideways. Thus he managed always to land at the end of the path of an L. The assumption is that one horse step is double a rider-step.

The movement of the chess aswa (horse) or aswagati or the knight-movement became a fascinating study by itself for intelligent persons left to their own devices to pass their time during gaps between their busy preoccupations and has stayed that way till today. Thus it is that you find puzzles and quizzes on knight movement popping up in all sorts of places and circumstances even now. From the computer science Department of the University of Pennsylvania at Philadelphia, the energetic graduate student Gaurav Chakravorty writes in his personal blog called Tech Interview Puzzles that it is not possible for the knight to start at one corner of a chessboard, visit every other square just once and end up at the opposite corner. This is because the length of the path is odd and therefore the final square should be of a color different from the beginning square, he points out. A not too recent posting at a popular problems and puzzles site, WU riddles requires you to write a program for a knight to visit all the squares in a standard chessboard so that he lands finally at a square adjacent to the starting square (not diagonally). Through a query in their regular puzzles column, asked the readers to determine the number of moves a knight can make from the centre square of any one of the sides of a 3×3 section of a chessboard till it returns to the square it started from. The answer is 8, and the additional information is that it will not land in the middle square of the section. The 3 by 3 diagonal magic square is said by many sources to have been known in China about 2200 b.c.. [Claude Berge, Principles of Combinatorics, Academic Press 1971, cites F. M. Müller, Sacred Books of the East, vol.XVI (The Yi-King),Oxford University Press 1882.] The square is usually presented in the orientation shown here, but of course by rotation and reflection (but keeping the number symbols the right way up!) it can appear in eight different orientations.

4 9 2
3 5 7
8 1 6

If the successive numbers are joined by straight lines the result is a tour of the 3 by 3 board by moves with coordinates {0,1}, {1,1} and {1,2}. This tour is a geometrical pattern which is centrosymmetric, i.e. unaltered by 180° rotation. In Variant Chess (see The BCVS Variant Chess Pages), pieces with the moves {0,1}, {1,1} and {1,2} are currently known as wazir, fers and knight. The knight’s move, which some take to be the defining element of chess, may perhaps have first been seen in the path followed by the successive numbers in the 3×3 magic square. A combined wazir + fers + knight is known as a centaur. So the 3 by 3 magic square can be described as a centaur tour of the 3×3 board.

Professor R. Vasantha has pointed out that Mathematics owes many interesting problems to the game of chess. Her view seems to be amply vindicated from the interest shown by scholars on knight-moves alone. Mathematicians being different from other scientists in their love of abstractions, it is not surprising that tricks, puzzles and number patterns have always held their attention. The strange move of the knight in chess makes plenty of room for fascinating puzzle-setting. The knight is allowed to occupy any unoccupied space on the board, which is two columns and one row or two rows and one column away from the cell he is in, regardless of whether or not the intervening cells are occupied. The attempt to cover all the squares of the chess board with a knight’s move, without going over the same square a second time, must be as old as the invention of game itself. Even in Europe, as early as in the 18th century, Euler had worked out several closed and open solutions to a number of knight’s tour puzzles. Much of the information on the mediaeval chessmen tours comes from H. J. R. Murray‘s       A History of Chess (1913) and other manuscripts that he left unpublished at his death in 1955, particularly an incomplete one with the similar title, The Early History of the Knight’s Tour, written about 1930. These manuscripts are now kept at the Bodleian Library, Oxford University. The earlier works of Antonius van der Linde, Geschichte und Litteratur des Schachspiels 1874 and Quellenstudien zur Geschichte des Schachspiels 1881, are also important sources. The 1725 edition of Ozanam’s book of mathematical recreations, which is referred to by many historians has a number of entries on this subject as well.

In his History of Chess (1913) H.J.R.Murray describes an arabic manuscript, at that time #59 in the John Rylands Library, Manchester, scribed by Abu Zakariya Yahya ben Ibrahim al-Hakim, with the title Nuzhat al-arbab al-‘aqulfi’sh-shatranj al-manqul { The delight of the intelligent, a description of chess }. This contains two 8 by 8 tours, one by Ali C. Mani, an otherwise unknown chess player, and the other by al-Adli ar-Rumi, who flourished around 840 and is known to have written a book on Shatranj (the form of chess then popular), which however survives only in extracts in this and other manuscripts. In competition to the idea of this being the oldest, is the claim of an early Indian knight’s tour given on a half-chessboard in the verse work Kavyalankara by the Kashmirian poet Rudrata, ascribed to the reign of Sankaravarman, 884 – 903, the tour itself being believed to have been designed very much earlier.



34 47 22 11 36 49 24 1
21 10 35 48 23 12 37 50
46 33 64 55 38 25 2 13
9 20 61 58 63 54 51 26
32 45 56 53 60 39 14 3
19 8 59 62 57 52 27 40
44 31 6 17 42 29 4 15
7 18 43 30 5 16 41 28


60 11 56 7 54 3 42 1
57 8 59 62 31 64 53 4
12 61 10 55 6 41 2 43
9 58 13 32 63 30 5 52
34 17 36 23 40 27 44 29
37 14 33 30 47 22 51 26
18 35 16 39 24 49 28 45
15 38 19 48 28 26 25 50

H.J.R. Murray too was of the opinion that Rudrata’s half-board tour was the earliest known knight’s tour, but by 1930 his view had changed in favour of the above tour by Adli. However, it has to be noted that the Islamic Empire was at its height around 800, extending from North Africa to northern India, and its capital Baghdad was probably the largest city in the world at the time. The discovery, that the knight could make such a tour of the chessboard, wherever it may have been made at first, would soon have made its way to the capital. Rudrata’s knight tour can be repeated on the lower half of the board so as to give a complete though not closed tour. A diagram showing it in this form, reflected left to right, appears in “a Persian manuscript of the early 19th century probably compiled in northern India”.

Rudrata 1) Half-board tour repeated also on the lower side and both the tours connected.

1 30 9 20 3 24 11 26
16 19 2 29 10 27 4 23
31 8 17 14 21 6 25 12
18 15 32 7 28 13 22 5
33 62 41 52 35 56 43 58
48 51 34 61 42 59 36 55
63 40 49 46 53 38 57 44
56 47 64 39 60 45 54 37

Rudrata 2) left to right inverted half-board tour repeated on the lower side and both tours connected at a different appropriate point for continuity of knight passage.

1 30 9 20 3 24 11 26
16 19 2 29 10 27 4 23
31 8 17 14 21 6 25 12
18 15 32 7 28 13 22 5
35 54 45 60 33 64 47 50
44 57 34 53 46 49 40 63
55 36 59 4 61 38 51 48
58 43 56 37 52 41 62 39

A tour is reported in the Bhagavantabhaskara, described by Murray (1913, 1930) as a Sanskrit work on ritual, law and politics, written “either about 1600 or 1700. This tour first became known in the west in an article by Monneron (1776) written after the work of Euler. The Nilakantha tour is given three times in his book, numbered from different points, the first being attributed to King Sri Sinhana of Sinhaladvipa (now Sri Lanka), the second to Nilakantha’s father Samkara, and the third to Nilakantha himself. Murray reports that the tour is also given in the Sardarnama, a modern Persian chess work written in the Indian Deccan, 1796-8, by Shir Muhammad-Khan. The Harikrishna manuscript of 1871, reproduced in S. R. Iyer’s Indian Chess (1982), also contains the three versions of the Nilakantha tour, following others attributed to the Rajah Krishnaraj Wadiar of Mysore which date from 1852–68.

The late Krishnaraja Wodeyaru III (the proper Kannada way of spelling his name in English), King of Mysore, whose remarkable compendium Sri Tattva Nidhi consists of precious nuggets of information on music, literature and different art forms borrowed from a number of sources apart from his own compositions both in Kannada and in Sanskrit, was a lover of puzzles as well. He was also bitten by the knight’s tour bug and he composed a number of chitrakavyas (acrostics) depicting aswagati (horse, namely knight movement). This king also did not restrict himself to the 8×8 square situation of a standard chess board or to the need for covering all the available squares. He specialized in depicting the sequence of knight movements either in very interesting geometric designs or through slokas. Some cells which are not to be covered by the knight when shaded provide interesting designs. A closed solution is invariably obtained. One such poser with such a puzzle was found by members of my family in the possession of the scholar and collector Dr. Maya of Mysore. This poser has the innocent form of Rajendra (the royal prince Arjuna) giving instructions to Krishna how to hold horse-reins in the grip of his teeth! The word play must have taken into account the fact that the royal author of the sloka was both Krishna and Rajendra! We also had the pleasure to have a look at a small portion of the king’s manuscripts kept with the priceless library of the Oriental Research Institute at Mysore. We found the king’s elaborate clues in Sanskrit/Kannada for four different solutions for the knight’s tour in 12×12 square situations. In one of them all 144 squares had to be covered. Lovely geometric designs depicted the squares that had to be ignored in the other three cases. Several other aswagati murals can be seen at Jayachamaraja Art gallery at Mysore and have been referred to by Professor Vasantha.

There have been poets who used aswagati in constructing chitraslokas. As an example we have the 929th sloka of the 1008-verse homage paid by Sri Vedanta Desika, who lived in the 13th and 14th centuries, to the padukas (sandals) of Sriranganatha, the presiding deity of the Srirangam Shrine, called Sri Padukasahasram. When the 32 aksharas of this sloka are arranged serially in the 32 squares of one half of the chess board, we are able to construct the next, namely 930th sloka, simply by starting from the akshara in the first square and moving on to pick up subsequent aksharas by successive aswagati movements! All forty slokas namely slokas numbered 911 to 950 in this masterly kavya are chitraslokas of one kind or other and the chapter holding these verses is appropriately called chitra paddhati by the author.

It is appropriate to end this article with a poser for those of its readers, who would still be awake at the end of the previous paragraph. I refer them to the WU riddle described in the fourth paragraph of the article, where the reader is asked to design a knight’s tour in a standard 8×8 chess board so that the knight arrives at the end of his tour of all squares at a square adjacent to the starting square ( not diagonally adjacent). Good luck!

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