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It is known that there are 1728 open knight's tours on a $5\times5$ chessboard.

Now I want to apply that to a very particular $5\times5$ square known as Sator square:

S A T O R
A R E P O
T E N E T
O P E R A
R O T A S

If we take letters from this square in the order of the knight's moves, every knight's tour will result in some 25-letter "word", however, many of these words will be the same because of the intrinsic symmetries of the Sator square. Note that none of the words can start or end with a vowel.

How many distinct 25-letter words will result?

Is there a way to come up with their number other than by brute force counting?

Vosoni
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    Given that the open knights tour is essentially counted through brute force computer algorithms, why would you expect your approach (which doesn't have an ideal symmetry because of repeated letters) to have a cleaner approach? – Calvin Lin Apr 02 '25 at 14:23
  • I assume I already know the result of the knight's tours. So if the letters didn't repeat, I suppose the answer would be $304\times2+56\times3+64=840$. However, because the letter R is repeating, some of the words starting with RE... could still coincide. – Vosoni Apr 02 '25 at 14:31
  • BTW, while the knight's tours starting from vowels cannot cover the whole square, in the original Sator square such a tour forms a meaningful sentence: ORO TE, PATER, which is probably intentional [Source: German Wikipedia]. – Vosoni Apr 02 '25 at 15:46
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    Maybe, you can also try to find many different "meaningful" solutions by considering a different leaper from Fairy Chess such as the Wazir (our last paper on the topic has been published today: https://nntdm.net/volume-31-2025/number-1/54-68/). – Marco Ripà Apr 02 '25 at 15:58
  • Nice idea. My motivation behind this was to make some puzzle by assigning different digits to the 8 letters and requiring that all knight's tours starting at the middle of the board are prime or something along this line. If that doesn't work with conventional knight's tours, trying different other leapers (e.g. from Omega chess) would be a way to try. The total number of ways to assign different digits to 8 letters is $10!/2=1814400$ (with zero) or $9!/2=181440$ (without zero). I just need to balance the number of constraints so that only one or few of these assignments satisfy the condition. – Vosoni Apr 02 '25 at 16:13
  • @MarcoRipà Looking at your paper, for the $5\times5$ Sator square only leapers with $a\leq2$ and $b\leq2$ would have a nonzero number of tours, of which all but Knight and Wazir stay on squares of the same color all the time. So actually there are no other alternatives than Wazir if one sticks to the requirement that the tour should cover the whole square and not end prematurely. Wazir is somehow degenerate, considering that it has nothing to leap over but just moves between adjacent squares. – Vosoni Apr 02 '25 at 16:29

1 Answers1

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I found the answer to my question. The number of distinct 25-letter words formed by knight tours on the Sator square is 276. These words are (in alphabetical order):

NAPATOROTAPATOROTERESERES, NAPATOROTAPATOROTESERESER, NATOROTAPATOROTAPERESERES, NATOROTAPATOROTAPESERESER, NOROTAPATOROTAPATERESERES, NOROTAPATOROTAPATESERESER, NOTAPATOROTAPATORERESERES, NOTAPATOROTAPATORESERESER, PANATOROTAPATOROTERESERES, PANATOROTAPATOROTESERESER, PATONATOROTAPATORERESERES, PATONATOROTAPATORESERESER, PATORONATOROTAPATERESERES, PATORONATOROTAPATESERESER, PATOROTANATOROTAPERESERES, PATOROTANATOROTAPESERESER, PATOROTAPANATOROTERESERES, PATOROTAPANATOROTESERESER, PATOROTAPATONATORERESERES, PATOROTAPATONATORESERESER, PATOROTAPATORONATERESERES, PATOROTAPATORONATESERESER, REPANATOROTAPATOROTESERES, REPATONATOROTAPATORESERES, REPATORONATOROTAPATESERES, REPATOROTAPANATOROTESERES, REPATOROTAPATONATORESERES, REPATOROTAPATORONATESERES, RERONOTAPATOROTAPATESERES, REROTANOTAPATOROTAPESERES, REROTAPANOTAPATOROTESERES, REROTAPATONOTAPATORESERES, REROTAPATORONOTAPATESERES, REROTAPATOROTANOTAPESERES, REROTAPATOROTAPANOTESERES, RESEPANATOROTAPATOROTESER, RESEPATONATOROTAPATORESER, RESEPATORONATOROTAPATESER, RESEPATOROTANATOROTAPESER, RESEPATOROTAPANATOROTESER, RESEPATOROTAPATONATORESER, RESEPATOROTAPATORONATESER, RESEREPANATOROTAPATOROTES, RESEREPATONATOROTAPATORES, RESEREPATORONATOROTAPATES, RESEREPATOROTANATOROTAPES, RESEREPATOROTAPANATOROTES, RESEREPATOROTAPATONATORES, RESEREPATOROTAPATORONATES, RESERERONOTAPATOROTAPATES, RESEREROTANOTAPATOROTAPES, RESEREROTAPANOTAPATOROTES, RESEREROTAPATORONOTAPATES, RESEREROTAPATOROTANOTAPES, RESEREROTAPATOROTAPANOTES, RESERESEPANATOROTAPATOROT, RESERESEPATONATOROTAPATOR, RESERESEPATORONATOROTAPAT, RESERESEPATOROTANATOROTAP, RESERESEPATOROTAPANATOROT, RESERESEPATOROTAPATONATOR, RESERESEPATOROTAPATORONAT, RESERESEPATOROTAPATOROTAN, RESERESERONOTAPATOROTAPAT, RESERESEROTANOTAPATOROTAP, RESERESEROTAPANOTAPATOROT, RESERESEROTAPATONOTAPATOR, RESERESEROTAPATORONOTAPAT, RESERESEROTAPATOROTANOTAP, RESERESEROTAPATOROTAPANOT, RESERESEROTAPATOROTAPATON, RESERESETANOROTAPATOROTAP, RESERESETAPANOROTAPATOROT, RESERESETAPATONOROTAPATOR, RESERESETAPATORONOROTAPAT, RESERESETAPATOROTANOROTAP, RESERESETAPATOROTAPANOROT, RESERESETAPATOROTAPATONOR, RESERESETAPATOROTAPATORON, RESERESETONAPATOROTAPATOR, RESERESETORONAPATOROTAPAT, RESERESETOROTANAPATOROTAP, RESERESETOROTAPANAPATOROT, RESERESETOROTAPATONAPATOR, RESERESETOROTAPATORONAPAT, RESERESETOROTAPATOROTANAP, RESERESETOROTAPATOROTAPAN, RESERETANOROTAPATOROTAPES, RESERETAPANOROTAPATOROTES, RESERETAPATONOROTAPATORES, RESERETAPATOROTANOROTAPES, RESERETAPATOROTAPANOROTES, RESERETAPATOROTAPATONORES, RESERETONAPATOROTAPATORES, RESERETORONAPATOROTAPATES, RESERETOROTANAPATOROTAPES, RESERETOROTAPATONAPATORES, RESERETOROTAPATORONAPATES, RESERETOROTAPATOROTANAPES, RESERONOTAPATOROTAPATESER, RESEROTANOTAPATOROTAPESER, RESEROTAPANOTAPATOROTESER, RESEROTAPATONOTAPATORESER, RESEROTAPATORONOTAPATESER, RESEROTAPATOROTANOTAPESER, RESEROTAPATOROTAPANOTESER, RESETANOROTAPATOROTAPESER, RESETAPATONOROTAPATORESER, RESETAPATORONOROTAPATESER, RESETAPATOROTANOROTAPESER, RESETAPATOROTAPATONORESER, RESETONAPATOROTAPATORESER, RESETOROTANAPATOROTAPESER, RESETOROTAPANAPATOROTESER, RESETOROTAPATONAPATORESER, RESETOROTAPATOROTANAPESER, RETANOROTAPATOROTAPESERES, RETAPANOROTAPATOROTESERES, RETAPATONOROTAPATORESERES, RETAPATOROTANOROTAPESERES, RETAPATOROTAPANOROTESERES, RETAPATOROTAPATONORESERES, RETONAPATOROTAPATORESERES, RETORONAPATOROTAPATESERES, RETOROTANAPATOROTAPESERES, RETOROTAPATONAPATORESERES, RETOROTAPATORONAPATESERES, RETOROTAPATOROTANAPESERES, RONOTAPATOROTAPATERESERES, RONOTAPATOROTAPATESERESER, ROTANOTAPATOROTAPERESERES, ROTANOTAPATOROTAPESERESER, ROTAPANOTAPATOROTERESERES, ROTAPANOTAPATOROTESERESER, ROTAPATONOTAPATORERESERES, ROTAPATONOTAPATORESERESER, ROTAPATORONOTAPATERESERES, ROTAPATORONOTAPATESERESER, ROTAPATOROTANOTAPERESERES, ROTAPATOROTANOTAPESERESER, ROTAPATOROTAPANOTERESERES, ROTAPATOROTAPANOTESERESER, SEPANATOROTAPATOROTERESER, SEPATONATOROTAPATORERESER, SEPATORONATOROTAPATERESER, SEPATOROTANATOROTAPERESER, SEPATOROTAPANATOROTERESER, SEPATOROTAPATONATORERESER, SEPATOROTAPATORONATERESER, SEREPANATOROTAPATOROTERES, SEREPATONATOROTAPATORERES, SEREPATORONATOROTAPATERES, SEREPATOROTANATOROTAPERES, SEREPATOROTAPANATOROTERES, SEREPATOROTAPATONATORERES, SEREPATOROTAPATORONATERES, SERERONOTAPATOROTAPATERES, SEREROTANOTAPATOROTAPERES, SEREROTAPANOTAPATOROTERES, SEREROTAPATONOTAPATORERES, SEREROTAPATORONOTAPATERES, SEREROTAPATOROTANOTAPERES, SEREROTAPATOROTAPANOTERES, SERESEPANATOROTAPATOROTER, SERESEPATONATOROTAPATORER, SERESEPATORONATOROTAPATER, SERESEPATOROTAPANATOROTER, SERESEPATOROTAPATONATORER, SERESEPATOROTAPATORONATER, SERESEREPANATOROTAPATOROT, SERESEREPATONATOROTAPATOR, SERESEREPATORONATOROTAPAT, SERESEREPATOROTANATOROTAP, SERESEREPATOROTAPANATOROT, SERESEREPATOROTAPATONATOR, SERESEREPATOROTAPATORONAT, SERESEREPATOROTAPATOROTAN, SERESERERONOTAPATOROTAPAT, SERESEREROTANOTAPATOROTAP, SERESEREROTAPANOTAPATOROT, SERESEREROTAPATONOTAPATOR, SERESEREROTAPATORONOTAPAT, SERESEREROTAPATOROTANOTAP, SERESEREROTAPATOROTAPANOT, SERESEREROTAPATOROTAPATON, SERESERETANOROTAPATOROTAP, SERESERETAPANOROTAPATOROT, SERESERETAPATONOROTAPATOR, SERESERETAPATORONOROTAPAT, SERESERETAPATOROTANOROTAP, SERESERETAPATOROTAPANOROT, SERESERETAPATOROTAPATONOR, SERESERETAPATOROTAPATORON, SERESERETONAPATOROTAPATOR, SERESERETORONAPATOROTAPAT, SERESERETOROTANAPATOROTAP, SERESERETOROTAPANAPATOROT, SERESERETOROTAPATONAPATOR, SERESERETOROTAPATORONAPAT, SERESERETOROTAPATOROTANAP, SERESERETOROTAPATOROTAPAN, SERESERONOTAPATOROTAPATER, SERESEROTANOTAPATOROTAPER, SERESEROTAPANOTAPATOROTER, SERESEROTAPATONOTAPATORER, SERESEROTAPATORONOTAPATER, SERESEROTAPATOROTANOTAPER, SERESEROTAPATOROTAPANOTER, SERESETANOROTAPATOROTAPER, SERESETAPANOROTAPATOROTER, SERESETAPATONOROTAPATORER, SERESETAPATOROTANOROTAPER, SERESETAPATOROTAPANOROTER, SERESETAPATOROTAPATONORER, SERESETONAPATOROTAPATORER, SERESETORONAPATOROTAPATER, SERESETOROTANAPATOROTAPER, SERESETOROTAPATONAPATORER, SERESETOROTAPATORONAPATER, SERESETOROTAPATOROTANAPER, SERETANOROTAPATOROTAPERES, SERETAPATONOROTAPATORERES, SERETAPATORONOROTAPATERES, SERETAPATOROTANOROTAPERES, SERETAPATOROTAPATONORERES, SERETONAPATOROTAPATORERES, SERETOROTANAPATOROTAPERES, SERETOROTAPANAPATOROTERES, SERETOROTAPATONAPATORERES, SERETOROTAPATOROTANAPERES, SERONOTAPATOROTAPATERESER, SEROTANOTAPATOROTAPERESER, SEROTAPANOTAPATOROTERESER, SEROTAPATORONOTAPATERESER, SEROTAPATOROTANOTAPERESER, SEROTAPATOROTAPANOTERESER, SETANOROTAPATOROTAPERESER, SETAPANOROTAPATOROTERESER, SETAPATONOROTAPATORERESER, SETAPATOROTANOROTAPERESER, SETAPATOROTAPANOROTERESER, SETAPATOROTAPATONORERESER, SETONAPATOROTAPATORERESER, SETORONAPATOROTAPATERESER, SETOROTANAPATOROTAPERESER, SETOROTAPATONAPATORERESER, SETOROTAPATORONAPATERESER, SETOROTAPATOROTANAPERESER, TANOROTAPATOROTAPERESERES, TANOROTAPATOROTAPESERESER, TAPANOROTAPATOROTERESERES, TAPANOROTAPATOROTESERESER, TAPATONOROTAPATORERESERES, TAPATONOROTAPATORESERESER, TAPATORONOROTAPATERESERES, TAPATORONOROTAPATESERESER, TAPATOROTANOROTAPERESERES, TAPATOROTANOROTAPESERESER, TAPATOROTAPANOROTERESERES, TAPATOROTAPANOROTESERESER, TAPATOROTAPATONORERESERES, TAPATOROTAPATONORESERESER, TONAPATOROTAPATORERESERES, TONAPATOROTAPATORESERESER, TORONAPATOROTAPATERESERES, TORONAPATOROTAPATESERESER, TOROTANAPATOROTAPERESERES, TOROTANAPATOROTAPESERESER, TOROTAPANAPATOROTERESERES, TOROTAPANAPATOROTESERESER, TOROTAPATONAPATORERESERES, TOROTAPATONAPATORESERESER, TOROTAPATORONAPATERESERES, TOROTAPATORONAPATESERESER, TOROTAPATOROTANAPERESERES, TOROTAPATOROTANAPESERESER.

Vosoni
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    Your first several tours start with RA, but I don't see how you can get from R to A with a single knight move. Or am I missing something? – paw88789 Apr 25 '25 at 21:40
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    You are perfectly right, there was a stupid mistake in my code. Thanks for spotting this. I have updated my answer. – Vosoni Apr 28 '25 at 07:14