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World’s oldest Combinatoric Formula


Pronounced as yamaataaraajabhaanasalagam and written in sanskrit as ???????????????

This is a word used extensively throughout the ancient Sanskrit grammar works. The greatness of this word lies in the fact that

  • This is the world’s oldest Combinatoric formula
  • This is the world’s oldest known de Bruijn Sequence
  • This is the world’s oldest known ‘Shift Register’
  • This is one of the world’s oldest known memory wheel or mnemonic (because there are many other such memory wheels in ancient vedas)

First, let us see what this Sanskrit word represents. It represents a binary sequence as follows:

Consider the word ‘Canada’. This can be split into Ca-na-da where all three syllables require the same amount of time to pronounce.
Now consider the word ‘America’. This can be split into A-me-ri-ca where A-ri-ca require the same amount of time to pronounce where as ‘me’ in America requires twice the amount of time as ‘A’ or ‘ri’ or ‘ca’ to pronounce.

So in Sanskrit grammar we have short syllables (called Laghu) and long syllables (called Guru). Examples of laghu above are Ca,Na,Da in Canada and A,ri,ca in America. Example of Guru is the long syllable Me in America.

Short syllables are denoted by 1 and long syllables by 0. So we can write the syllables of the above mentioned sanskrit word ‘yamatarajabhanasalagam’ as 1000101110

Now let us see how this is a Combinatoric. Combinatorics is all about combinations. An example is, given 9 unique symbols, in how many unique ways can we pick a group of 3 symbols from it?

A second category of problems is a more tougher version of the above problem, given 3 unique symbols in how many unique ways can we pick a group of 9 symbols from it, by allowing repeated pick of a given symbol?

‘yamatarajabhanasalagam’ as a combinatoric represents a solution to one such problem of the second type where we have two symbols 0 and 1 and we have to find out as to how many unique groups of three can we pick up from it by allowing repeated picks?

The simple answer to the problem here is just divide the given formula into groups of three by shifting one place at a time as follows:

1000101110 = 100,000,001,010,101,011,111,110

Now this represents all the 8 possible combinations for the above mentioned problem of arranging 0 and 1 into groups of three! In other words, this is the list of all possible triplets of a binary sequence!

A Shift Register too!

Since the solution is in the form of shifting one place at a time from the left, this is also the world’s first ’shift register’! Note that shift registers are used extensively in modern computers to speed up calculations. Software programmers might be aware of left shifts and right shifts << and >>


The main purpose of this word is to use it as a mnemonic or a memory wheel. That would make it easy for one to remember and quickly recall all possible combinations. There are hundreds of such words that are used in Sanskrit as mnemonics to help people memorize mathematical numbers and formulae. The authors of these mnemonics have been so creative that they have created sacred hymns, short stories etc which initially look like genuine hymns or short stories or sentences or proverbs or riddles etc, but when you decrypt them into numbers you end up with a hashing algorithm , or the value of PI to infinite decimal places, or with a logarithm etc, or with a formula etc…

The first memory wheel in modern history of mathematics appears only in 1882 where one such memory wheel was created by the French Mathematician Emile Baudot!! Can you imagine how advanced ancient Indian grammar and mathematics was!! Sometimes I feel we are actually living in a technologically inferior era compared to ancient Indians where today we are just reinventing the wheel !!

De Bruijn Sequence

These are special types of sequences first studied in modern history by de Bruijn and are defined as ‘Given a set S of words of length n, a de Bruijn sequence of span n is a periodic sequence such that every word in S (and no other n-tuple) occurs exactly once.’ In simple terms, de Bruijn sequences are nothing but the shift registers mentioned above!

Where is ‘yamAtArAjabhAnasalagam’ used?

In Sanskrit grammar this is used to divide poetry into a collection of three syllables called Ganas. ‘Yamatarajabhanasalagam’ defines all the 8 possible Ganas as follows

  • ‘Ya’ Gana is 100
  • ‘Ma’ Gana is 000
  • ‘Ta’ Gana is 001
  • ‘Ra’ Gana is 010
  • ‘Ja’ Gana is 101
  • ‘Ba’ Gana is 011
  • ‘Na’ Gana is 111
  • ‘Sa’ Gana is 110

The gana combinations are then used to define the rules for writing poetry. It is rules like these that form the basis of the most mathematical and scientific human spoken language – Sanskrit, which is why it was termed to be the only human spoken language with the ability to become a software programming language because of its precision – the research was done by the Forbes Magazine of Germany.

Which is why I have always felt that this language was not born on earth, but instead is of an alien origin, the language spoken by aliens with advanced technology and science. Those aliens were probably Type II or Type III civilizations!! Note that ancient vedic texts call Sanskrit as ‘Deva Bhaasha’ which means ‘A language of Divine (alien?) origin’

Well, divine Indeed !

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Gurudev is the developer of Gurunudi AI Platform. This is his official website where he pens his thoughts on a wide range of topics, answers queries, shares resources and tools developed by him.


  1. Sainath
    Its used everywhere in sanskrit grammar.. called Chandhas.
    These are taught in the language classes in schools as part of grammar and can be found even in languages derived from Sanskrit like Kannada, Telugu, Hindi etc. I remember these things from my Kannada classes.

    Chandas, Vrtta, Lakshana all these will have to be understood with the poetic meters used, with the basis being a binary system of short and long syllables called laghu (1) and guru (0)

    In music, the beat combinations are remembered using the 10 fingers to represent the memory wheel yam?t?r?jabh?nasalagam, where in left hand thumb and small finger are bent, and in the right hand the three middle fingers are bent, so all bent fingers represent guru, and all other fingers represent laghu, and then one can practice the beats.

  2. i am sorry gurudev, but if you can provide some ex. it would be good as to understand how exatly this was used.
    sorry for the trouble…

  3. Its used in ancient sanskrit grammar and while teaching the students ancient Indian classical music, to help them easily remember all the basic combinations of rhythm!

  4. okie let me give another example..
    consider the word rajasthan
    here “”ra”” is twice as long as “”ja””
    I mean “”ra’ is actually pronounced as “”raa””, and “”ja”” as “”ja”” itself,
    so “”ra”” is guru (long one) and “”ja”” is laghu (short one)!
    Hope you got it.

  5. As per Amitabh Bacchan’s dialogue in a movie – inglish ees a phunni langvaje it may be difficult to identify short/long syllable. I am just providing consonent with actual list of vowels. Nowadays few vowels are dropped out in educational text books itself.

    Here is extract of Baraha – a trasliteration free s/w help on consonent with Vowels . Hope it helps the interested

    ka kA ki kI ku kU kRu kRU klRu klRU k~e ? ?? ?? ?? ?? ?? ?? ?? ???? ???? ??

    ke(short ‘e’) kE kai k~o ko(short “”o””) kO kau ?? ?? ?? ?? ?? ?? ??

    kaM kaH
    ?? ??

    According to the pronouciation and above scripting America in different indian language would be
    1 0 1 ? ( is it akaarantha masculine gender word)
    a mE ri ka
    ? ?? ?? ? – Devanagari/Hindi
    ? ?? ?? ? – Kannada
    ? ?? ?? ? – Malayalam
    ? ?? ?? ? – Tamil
    ? ?? ?? ? – Telugu
    ? ?? ?? ? – Bengali

  6. No, you are wrong… gam is a guru because of the anusvara with it… by definition of a guru.. gam and gAm both are guru

    So it is
    yamAtArAjabhAnasalagam = 1000101110
    yamAtArAjabhAnasalagAm = 1000101110
    And either way the formula is correct !

  7. Some say yamAtArAjabhAnasalagAm is the original version while some say yamAtArAjabhAnasalagam is the original version. In modern days yamAtArAjabhAnasalagAm is taught as the sutra..
    Another point to be noted is that in the sutra the first 8 letters refer to the formula, and the last two letters refer to la=laghu and ga=guru..
    So in that sense, I think it is gam and not gAm

  8. ?Atyatistaddashangulam? is not that simple… dasha angulam is not few inches.. dasha is 10 and angulam here refers to dimensions not inches :)
    This line is from PurushaSookta.. Purusha is the universal soul whose unification with the Prakriti (matter-energy) creates the physical universe we live in…
    It says that the universal soul is beyond the 10 dimensions of the physical universe.. note that only modern string theory says that the universe has 10 dimensions..
    But the vedic text Vishnu Purana even has names to each of this 10 dimensions and even describes them all…
    I think I HAVE to write a post on it now :)

  9. about “”Atyatistaddashangulam”” i asked my teacher (high school ) and he said nothing more than “” several rounds to earth and few inches more “”

    There was nationalist magazine used to come called “”aseema”” , and that had this as its punchline.

  10. @bachodi

    Yes, ?ekadhikenapurvena? is the vedic sutra used for testing divisibility by primes, for 5 specific multiplication, etc..

    “”Atyatistaddashangulam”” as you said is not related to maths.. It is much more than that, it is related to Physics, in particular to the total number of dimensions of the universe!! Thanks for reminding me about this, will write a post on it soon :)

  11. Wonderful ,
    this reminds me one of the vedic mathematics line “”ekadhikenapurvena”” .. This is the onlything i remember among rules i learned.

    Do you have any reference for a sentence “”atyatistaddashangulam”” nothing to do with maths. But do you have any ?


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