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The knight's tour problem

The problem is having a knight traverse across all squares on a chessboard without visiting any square twice. The Padhuka Sahasram by Sri Vedanta Desikan presents a solution to the problem in verses 929 & 930. The first verse is written sequentially and the next is read along the path taken by the knight yielding the result.

929. sThirAgasAm sadhArADhyA vihathAkathathAmathA
sathpAdhukE! sarAsA mA rangarAjapadham naya

930. sThithA samayarAjathpA gatharA mAdhakE gavi
dhuranhasAm samnathAdhA sAdhyAthApakarAsarA

sthi
1
rA
30
ga
9
sAm
20
sa
3
dhA
24
rA
11
dhyA
26
vi
16
ha
19
thA
2
ka
29
tha
10
thA
27
ma
4
thA
23
sa
31
thpA
8
dhu
17
kE
14
sa
21
rA
6
sA
25
mA
12
ran
18
ga
15
rA
32
ja
7
pa
28
dha
13
mna
22
ya
5
1/
33

Translation by U.Ve. Dr. V.N. Vedanta Desikan Swamin:

Oh Paaduka of the Lord who is meant by the word 'Sat'! The worst of 
sinners can worship You and benefit. You ward off all evil 
happenings, which are inherently grief-causing. What a melodious walk 
You possess! You diligently keep guard in respect of people who stick 
to the righteous path with pure activities. You remain in an 
effulgent site with SRI, unique to You, granting affluence to the 
worshippers. You remove the pitiable condition that accrues to 
sinners. You make the strolls pleasing and relaxing to the Lord, when 
He has to go out. May You please lead me on to the Lord Ranganatha's 
Abode!

Special Notes by V.Sadagopan(V.S):

SlOkam 929 and 930

These two slOkmas are to be read together as one set.
This chithra bhandham is known as " Chathuranga-Turanga 
padha bhandham ". Chathurangam means a chariot and 
Turangam means a horse. These two slOkams fit like 
a horse-drawn carriage as a unit. The verse moves like 
the steps of a horse drawing the chariot . The geometric 
representation of the first verse and its relationship (symmetry)
to the subsequent verse shows the Turanga Padha kramam .
The 32 aksharams have to be presented in a 8x4 mode 
to appreciate this kramam (sequence).

There are 32 aksharams in the slOkam 929 as well as 
in sLokam 930. One has to position the 32 aksharams of
the first slOakam in four rows and follow it up with the 32
aksharams of the second slOkam in the same manner.
Then one has to use the movement regulations of a horse 
in the chess game ( chathurangam ). That movement is 
to jump to the next position and from there gallop to
the end of the diagonal posiiton .. 

1) UtthamUr Swamy's anubhavam( SlOkams 929 & 930): 
Oh Holy Paadhukais of the Lord ! Please bring me close to 
Lord RanganAthA's sacred feet ! You are the pair to be worshipped 
always by those burdened down by their bundles of sins .
You destroy their sorrows . You are celebrated for that blessing . 
You have a delectable sound , when You move about.
You protect the righteous. You are rich in wealth and attributes. 
You reside inside the rays that produce joy. You destroy the sorrow 
of those , who have committed heinous sins .Your movements 
banish the afflictions suffered by the above . May You bring Your 
Lord's feet to me or take me to where they are !

2) SrImath Andavan's anubhavam(SlOkam 929) : Oh Sath PadhuE ! 
You are the one fit to be worshipped by SamsAris in this world and 
to be recognized as the PurushArTam in the other world (during 
the stage of Mukthi) . You accept the Prapathti performed to You 
and destroy sanchitha and PrArabdha karmAs of the prapannAs 
and grant them Moksham . My prayers to you are to bless me
quickly with the attainment of the access to Your dear Lord's 
sacred feet in Parama Padham .

(3) SlOkam 929/ Oh Sath-PaadhukE ! ( the Paadhukais of Parabrahmam )!
SamsAris have unquenchable sins ( sTira aagasAm) . For such
nithya samsAris , You are the fit object of worship ( sTira aagasAm
sadhA aarAdhyA ). Oh glorious Paadhukais which remove 
inauspiciousness ( viahatha aka tathaa amathaa ) and have 
the pleasing naadham ( saraasA ) ! Please lead me to 
the sacred feet of Your Lord ( Maa RangarAja padham naya) . 

4) SrImath Andabvan's anubhavam: (SlOkam 930) :The PaadhukAs have 
the following attributes and the prayer to the Paadhukais is to connect
the samsAri to the sacred feet of the Lord at SrI Vaikuntam :

(a) It protects the righteous , who do not slip from their sath-
SampradhAyam through the power of their austerity .

(b) It has all the Iswaryams under its control and distributes them
according to its desires to aasrithAs .

(c) It becomes a fit object for dhyAnam as it sits in the middle of
its brilliant rays as a JyOthirmaya vasthu that gladdens the 
heart of all beholders .

(d) It removes all the fierce sins of samsAris , the moment they pray
to it for protection through the contact with its lustrous rays .

(e) It has the power to travel to all corners of the world to banish
the sorrows of those , who seek its protection . 

5) The sacred PaadhukAs of the Lord protect those who shine 
with their observance of the Sath-sampradhAya anushtAnams 
and ask those PaadhukAs to lead Swamy Desikan to
the sacred feet of Lord RanganAtha (Samaya Raajath Paa ThvAm
Maa RangarAja Padham naya ) . Other attributes of the PaadhukAs 
qualifying them to lead Swamy Desikan to the sacred feet of the Lord 
of SrIrangam are next described : (a) It is the possessor of parama 
Iswaryam ( aagatha raa ) (b) it stays in the middle of Soorya MaNDalam
in an auspicious state resplendent with its own jyOthi ( MaadhakE gavi 
sTithA), (c) It destroys the fierce sins of samsAris and grants them 
an auspicious state ( dhuramhasAm sannathA dhaa) , (d) it has 
the tranquil and cool rays that travel everywhere ( Saadhya atApa kara 
aasArA). May Thou lead me to the closeness of Your Lord ( Thvam 
Maa RangarAja paadham naya )...(V.S).