This page is devoted to many contributions to Combinatorics from Indian Classics (currently containing classics in Sanskrit). It shows various solutions to the Knights tour problem - that of making a Chess Knight visit all squares of an chessboard exactly once. This problem has fascinated many for centuries, including the great mathematican Euler. The solutions on this page are for a 4x8 board, but can be seamlessly repeated to get a complete solution for the whole board. They include both a prayer form in the "Padukaa Sahasra" about 1300 A.D., as well as the earlier forms in the HaraVijaya of Ratnakara (about 800 A.D.), and the Kavyalankaraa by Rudrata (about 900 A.D.). The links on the page are to various Sanskrit classics, which have information regarding both Knight's tours, as well as other amazing combinatorial jugglery with Sanskrit poems.

The pictures below show the Knights tour in the Padukaa Sahasra, a piece with 1008 verses devoted to Lord SitaRama's sandals (worshipped by Bharata during Lord SitaRama's stay away from Ayodhya), written by Sri Vedanta Desikar around 1300 AD. These tours are presented in Chitra-Kavya form (picture-poem in Sanskrit). This poetic form begins with one verse, and derives another meaningful one, by systematic permutations (e.g. according to a Knight's tour of a board) of the same - an amazing feat! Several more Chitra-Kavya's are shown in the same Padukaa Sahasra page - e.g. "LokaTaraaKamaChara ...", is written half on a chessboard, reflected through the center, and the result can be read sequentially either horizontally or vertically in a meaningful sense on the board.

(a)The JPEG file "board.jpg" shows the Knight's tour in the top left corner. The first sloka is written sequentially, and the second one is read off it, following the tour - which is also marked in Devanagari numerals.

(b) The JPEG file "slokas.jpg" shows the two slokas - the first on the top left of the left page, and the second following it

(c) The JPEG file "previous_sloka.jpg" lists the preceding Sanskrit material for reference only.

(d) The JPEG file "cover.jpg" is the cover page of the 1911 book from which we have taken this material. The translation of these slokas (following the commentator Sriniwasa) is given below:

Chess knight circuit (G. S. S. Murthy's Translation)

Sthiraagasaam sadaaraadhyaa vihataakatataamataa| Satpaaduke saraasaa maa rangaraajapadam naya||

Sthitaa samayaraajatpaagataraa modake gavi| Duramhasaam sannataadaa saadhyaataapakaraasaraa||

The author beseeches the sandals of God to help him reach the abode or the feet of God. The two slokas together form one sentence.

Sat+paaduke = O Sandals of Brahma,

Sthira+aagasaam, Sadaa+aaraadhyaa = (you are )always revered by the worldly who are confirmed sinners,

Vihata+aka+tata+amataa = (you are) the destroyer of unhappiness and instruments of unhappiness,

Saraasaa = (you are) always accompanied by (the jingling) sound (of trinket bells attached to the sandals?)

Maa, naya = take me,

Rangaraaja +padam = to the feet or abode of God.

Samaya+ raajat+paa = (you are ) the protector of those shining through their good conduct,

Aagata+raa = (you have) attained brilliance of gold,

Modake = (you) dispense joy,

Gavi, sthitaa = (you) are at the center of sun,

Durahmasaam, sannataa+daa = (you are) destroyer of the despondency of the wicked,

Saadhya+ataapa+karaa = rays (of your gems) have the power of removing the heat (of the devotee),

Aasaraa = (you) move all around,

English "Chitra-Kavya's" from Knuth (Jan 2005) for these two tours are also listed there.

One knight jumps like three rook-wise steps.

Past sore too mean; so just for free,

Hopes here, turns there, flies each goose now.

Can't place last word? Won't find the sea

One, two, three, four! See each word here:

Jumps so wise now find their place passed.

Terns can't soar like flies the free rook;

Goose steps just won't mean knight hops last.

- D. E. Knuth, Jan 6, 2005

Similar Knight's tours are in the Kavyalankaraa by Rudratra (around 900 A.D.) and in the Haravijaya by Ratnakara
(830 A.D. according to David Smith of Oxford). The - tour in the Haravijaya yield's a complete Hamilton cycle (Knuth - Jan 2005) and is probably the oldest tour known.

Excerpts on the various forms of Chitra-Kavya's and related poetics from Kunhan Raja's "Survey of Sanskrit Literature" Bharatiya Vidya Bhavan, 1962 are also included.

For reference we have scanned and listed the relevant pages of

(1) The Paduka Sahasra of Vedanta Desikar,
    Slokas P1- P2- P3- P4- P5- P6- P7- P8- P9- P10- P11- P12- P13- P14
    Pictures P1- P2- P3- P4- P5
(2) The Haravijaya of Ratnakara
    P1
(3)The KavyaDarsha of Dandin
    P1- P2- P3- P4- P5- P6- P7- P8- P9- P10- P11- P12- P13- P14
(4) The Kavyaalankara of Rudrata
    Slokas P1- P2- P3- P4- P5- P6- P7- P8- P9- P10- P11- P12- P13- P14- P15- P16- P17- P18- P19- P20- P21- P22- P23- P24- P25- P26- P27- P28- P29- P30
    Pictures P1- P2- P3- P4- P5- P6- P7- P8
They can be downloaded. The file names used are sequential with respect to page number in the text, and should be self-explanatory - try them in a JPEG viewer. More material will be added on a continual basis.

We thank the publishers - Parimal Publications, New Delhi  and Chowkhambha Publications for permission to use this material.

More material will be added to this site on a continual basis. If any of you have information related to these topics, please drop us an email at gnsprasanna@iiitb.ac.in