## Story of how the young Buddha passes his maths test

King Suddhodana then asked the
Bodhisattva:
“Can you, my son, rival the skill of the great mathematician Arjuna
in the knowledge of mathematics?”

“Sire, I can,” he replied.
So the Bodhisattva was told to show his ability.

### §1. The Bodhisattva knows up to ten numerations

The great mathematician Arjuna asked the Bodhisattva:
“Young man, do you know the procedure of numeration called
kotisatottara,
more than a hundred kotis?”

The Bodhisattva answered: “I do.”
[usually a koti
is ten million or `10^7`

]

“Well then, how must one proceed
to enumerate more than a hundred kotis?”

The Bodhisattva replied:
“A hundred kotis is called ayuta;
a hundred ayutas is called niyuta;
a hundred niyutas called kankara;
a hundred kankaras is called vivara;
and a hundred vivaras is called aksobhya;
a hundred aksobhyas is called vivaha;
a hundred vivahas is called utsanga;
a hundred utsangas is called bahula;
a hundred bahulas is
called nagabala;
a hundred nagabalas is called titila;
a hundred titilas is called vyavasthanaprajnapti;
a hundred vyavasthanaprajnaptis is called hetuhila;
and a hundred hetuhilas is called karahu;
a hundred karahus is called hetvindriya;
a hundred hetvindriyas is a samaptalambha;
a hundred samaptalambhas is known as gananagati;
a hundred gananagatis is called niravaravadya;
a hundred niravaravadyas is called mudrabala;
a hundred mudrabalas is called sarvabala;
and a hundred sarvabalas is called visamjnagati;
a hundred visamjnagatis is a sarvasamjna;
a hundred sarvasamjnas is a vibhutangama;
and a hundred vibhutangamas is called tallaksana.
[then `100^23`

kotis would mark `10^53`

]

“Now with the numeration called
tallaksana
one could take even Meru, the king of mountains,
as a subject of calculation and measure it.
And next is the numeration called
dvajagravati;
with the help of this numeration one could take all the sands
of the river Ganges as a subject of calculation and measure them.

Above this is the numeration called
dvajagranisamani;
and above this is the numeration of
vahanaprajnapti;
next comes the numeration called
inga;
above this is the numeration of
kuruta.

Again above this is the numeration called
sarvaniksepa,
with the help of which one could take the sands of ten Ganges rivers
as a subject for calculation and measure them all.

And again above this is the numeration called
agrasara,
with the help of which one could take the sands
of a hundred kotis of Ganges rivers as a subject of calculation measure them all.
[these sands of the Gangeses
would be too few for their respective numerations]

“And again above this is the highest numeration called uttaraparamanurajahpravesa, which is said to penetrate the most subtle atoms. Except for a Tathagata, or a Bodhisattva who has reached the purest essence of Enlightenment, or a Bodhisattva who has been initiated into all the Dharma, there is no being who knows this numeration, except myself or a Bodhisattva like me, who has arrived at his last existence, but has not yet left home.”

### §2. Counting the atoms in a yojana and the Earth's mass

Arjuna said:
“Young man, how must one proceed in the numeration
which penetrates the dust of the most subtle atoms?”
[counting back this list of lengths an atom
would measure between `1`

and `1000`

picometer,
in reality its diameter is `60`

to `600`

pm]

The Bodhisattva said:
“Seven subtle atoms make a fine particle;
seven fine particles make a small particle;
seven small particles make a particle called
vatayanaraja;
and seven particles of vatayanaraja make a particle called
sasaraja;
seven particles of sasaraja make a particle called
edakaraja;
seven particles of edakaraja make a particle of
goraja;
seven particles of goraja make a liksaraja;
seven liksaraja make a sarsapa;
seven sarsapas make an adyava;
seven adyavas make an anguli;
twelve anguli make a parva;
two parva make a hasta;
four hastas make a dhanu;
a thousand dhanu make a
krosa of the country of Magadha;
four krosas make a
yojana.
[a yojana measures a day's march of a royal army in distance,
here covering about `108*10^12`

atoms]

And now who among you knows the mass of one yojana,
and how many of these subtle atoms it contains?”

Arjuna said: “I myself am even more astonished than others of lesser knowledge. Let the young prince show us the mass of a yojana, and explain how many subtle particles are found in it.”

The Bodhisattva replied:
“In the mass of a yojana
there are a complete niyuta of aksobhyas
plus thirty hundred thousand of niyutas of kotis
plus sixty hundreds of kotis plus
thirty-two kotis and five times a hundred thousand and twelve thousand.
[a 'mass' of `10003000000000000060320512000`

atoms?]
Such is the calculation of subtle particles in the mass of a yojana.

By this procedure, there are here in the land of
Jambu seven thousand yojanas;
in the land of Aparagodana, eight thousand yojanas;
in the land Purvavideha, nine thousand yojanas;
in the land of Uttarakuru, ten thousand yojanas.
[Earth `34000`

yojanas ~
måss `3.4E32`

atoms]

### §3. Three thousand great thousandfold world in essence incalculable

“Continuing with this method,
beginning with the worlds composed of four continents,
there are a hundred kotis of
worlds with four continents
and a hundred kotis of
great oceans;
there are the hundred kotis of
Cakravalas
and of Mahacakravalas;
the hundred kotis of Sumerus,
kings of mountains;
the hundred kotis of realms of the
Four Great Kings;
the hundred kotis of realms of the
Thirty-three gods;
the hundred kotis of realms of the
Yama gods;
the hundred kotis of
Tusita realms;
the hundred kotis of
Nirmanarata realms;
and the hundred kotis of
Parinirmita vasavartin realms.
There are the hundred kotis of
Brahma realms;
the hundred kotis of
Brahmapurohita realms;
the hundred kotis of
Brahmaparsadya realms;
the hundred kotis of
Mahabrahma realms;
the hundred kotis of
Parittabha realms;
the hundred kotis of
Apramanabha realms;
the hundred kotis of
Abhasvarana realms;
the hundred kotis of
Parittasubha realms;
the hundred kotis of
Apramanasubha realms;
the hundred kotis of Subhakrtsna realms;
the hundred kotis of
Anabhraka realms;
the hundred kotis of
Punyaprasava realms;
the hundred kotis of
Brhatphala realms;
the hundred kotis of Asangisattva realms;
the hundred kotis of Abrha realms;
the hundred kotis of Atapa realms;
the hundred kotis of Sudrsa realms;
the hundred kotis of Sudarsana realms;
and the hundred kotis of Akanistha realms.
[`3000`

kotis of worlds in total,
maybe `3*10^10`

, see commentary box]

“All together these are said to be the whole of the
three thousand great thousands of worlds,
spread out and developed.
All the calculations of the essence of the yojana
includes the many hundreds of yojanas of subtle particles
in this mass of three thousand great thousands of worlds,
the many thousands of yojanas,
the many kotis of yojanas,
and the many niyutas of yojanas.

And how many subtle particles are there?
It passes beyond calculation, it is
incalculable.
There are an incalculable number of subtle atoms
in the mass of the three thousand great thousands of worlds.”

### §4. Admiration of this mathematical lesson

While this lesson on enumeration was being taught by the Bodhisattva, the great mathematician Arjuna and the multitude of Sakyas listened with pleasure, joy, and happiness. Everyone there was filled with great admiration, and each of them presented the Bodhisattva with garments and ornaments. The great mathematician Arjuna then uttered these two verses:

“The hundreds of kotis and the ayutas,

the nayutas and the niyutas,

the procession of the kankaras, the vivahas,

and the aksobhyas as well:

this supreme knowledge I do not have – he is above me.

One with such knowledge of numbers is incomparable!

“And doubtless, O Sakyas, he could calculate

the dust of the three thousand worlds,

as well as all the herbs, the woods, the medicinal plants,

and even the drops of water,

in the time it takes to say ‘Hum’.

How could these five hundred Sakyas

do anything more wonderful?”

Then gods and men by the hundreds of thousands
uttered cries of admiration and joy.

And the
devaputras
in the expanse of the sky recited this verse:

“The concepts and the ideas,

the reasonings good or bad, small or great,

the workings of the minds of all the beings

of the three times: all this he knows perfectly

through a single movement of his mind.”

Thus, O monks, the Bodhisattva distinguished himself by his superiority over all the other young Sakyas. And as they continued their contests – in jumping, in swimming, in running and all the rest – the Bodhisattva again and again demonstrated his superiority...

## “Does it all add up in the sutras?”

by Asamkhyeya dasa

A commentary on the main points in the Lalitavistara Sutra's mathematical test of the young Buddha.

koti. It is doubtful if this number has the conventional value of`1E7`

, when we take into account the traditional number of worlds in athree thousand great thousandfold worldwhich is`1000^3 = 1E9`

. Here the list of worlds goes beyond the earthly and heavenly realms of desire (places of rebirth) and also covers the two spheres of form (or creation) and formlessness (or extinction), to add up to`3000`

kotisof worlds. This only makes sense if we put thekotivalue at`1E6`

.kotisat`1E8`

ends with atallakshanaat`1E52`

, a factor ten less than usual. Now thetallakshanamay be the first value of the next enumeration (also calledtallakshana) or the next numeration may start at a value after that. It's not clear if the multiplicative steps again should be`100`

and if the number of steps stays equal in all numerations, namely`22`

or`23`

. The upcoming comparisons with the sands of the Ganges seem to suggest these parameters become much smaller, but this can be brushed aside to keep the argument going at a reasonable pace. Most probably the Sutra writers didn't have a clue what they were doing, but their approach reminds us of the Big number algorithm of Archimedes and may be a flawed attempt to transplant his system to Indian buddhist soil.`8`

numerations starting from`1E52`

all have`22`

multiplicative steps of`100`

, then the last number expressed, theuttaraparamânurajahpraveša, is`10^(8+44*9) = 10^404`

or`100^202`

or`10000^101`

in which form it would have had great appeal to the Indian eye. Suddenly we have amyriadtimes amyriadand this repeated a hundred times, with a finalmyriadadded on top, beautiful! The right point to change over from mere mathematical numbers to the physical world – because curiously in the upcoming numeration of distances, this last number is made to express the smallest length of anatom. It is as if we were dealing with quantum-information that fills the vacuum here.`10`

th numeration of lengths goes from types of particles, to types of dust stirred up by animals of increasing size, to plant seeds, to human bodily measures, a mile, and then ending with the definition of ajoyana, the measure of distance most common in buddhist sutras. Thejoyanais a length of about`108*1E12`

of atoms in total,`108470495616000`

atoms to be precise, which can be traced back to a mathematical number (sic!) of about`1E418`

.joyana– with akotiof a million this måss equals`100030000000000006032512000`

atoms – cannot even cover the thinnest layer of atoms in the area of a squarejoyana(if akotiis`1E7`

it nearly does). But maybe we cannot believe the exact expression as given above, which certainly doesn't look like a serious number. Perhaps a few additional digits have been dropped during the translation or copying, perhaps the original writers were just fools trying to impress, or perhaps it was meant as a riddle. For now we faithfully trace this måss back to a number of about`1E430`

.`34000`

yojanas, to form an earth world of four continents, or more generally the unitworld. Such a world's måss again can be traced back to`3.4E30`

atoms and`3.4E434`

in pure number, which is what we have at the start of the enumeration of worlds that follows. There`100`

kotisof earth worlds (with akotiof`10^6`

) form a måss of about`3E38`

atoms or`3E442`

in pure number. In all`30`

types of realms are listed, each with a`100`

kotisof worlds, constituting thethree thousand great thousands of worlds. This seems to suggest their total can be traced back to a måss of`1E40`

atoms or`1E444`

in pure number, but there's a problem…samsâraworlds of rebirth). Even in our universe of desire there are the heavenly planets of thedevas, not likely composed of the earth-like particles described above as rabbit dust (šašaraja) and such. If these higher realms do consist of atoms, their type and number is bound to disagree with that of the lowly realm of earth. This being common buddhist knowledge it is mistake to think we can derive with certainty any total number from this list of worlds on top of its first entry: the earth worlds.We think the coming passage in the Lotus Sutra addresses just the problem left by the improbable definition of the

three thousand great thousands of worldsas a måss of atoms. If this is the case, the Lotus Sutra can be dated after the Lalitavistara Sutra.The issue here is the early evolution of the concept of a

three thousand great thousandfold world, culminating centuries later in its subdivision in theAbhidharma Koša, a commentary on buddhist doctrine attributed to Vasubhandu (5th century AD). There a`1000`

worlds form a small chiliocosm, there's a wall,`1000`

small chiliocosms form a middle chiliocosm, then again a wall, and this`1000`

times repeated to constitute the large chiliocosm. Apart from the convention to allow a total of a billion worlds, the distance between two worlds is settled there at`1203450`

joyana(about a light minute), a minor travesty.The Parable of the Magic City chapter in the Lotus Sutra starts with the Buddha explaining the time elapsed after the death of a former Buddha (called Universal Surpassing Wisdom). The argument is metaphorical, and builds further upon concepts we've met in the Lalitavistara above: the

three thousand great thousandfold world, the particles of dust, the country to the East, the reference to mathematicians.Precision is not what is aimed at here, the new quantities are just hinted at for their power to dazzle the mind, but actually the numbers do not become larger than in the Lalitavistara, as calculated within brackets. Important in this passage is that only the earth element is rightly qualified to be ground to ink.

Given the data from the Lalitavistara Sutra we solve the mathematical riddles in the Lotus Sutra as follows.

kotivalue is`1E7`

(the usual ten million) throughout, as we aim for maximal estimates.`100`

kotisof earth worlds in athree thousand great thousandfold world, the Buddha can have at most`1E9`

(a billion) worlds filled with earth.yojanacounts a length of`L`

atoms, and a måss of_{y}~1.08E14`M`

atoms._{y}~1E28To translate length

`L`

to måss`M`

a possible formula is`M`

(in buddhism so it seems).L^1.995~vâtâyanaraja(windy road dust), smallest dust in the list, has a diameter`L`

times the size of an atom. Then road dust can have a måss of_{v}=7^3`M`

times an atom's måss._{v}~7^6world with four continentscounts`34000`

yojanamåsses or`M`

atoms. Then we have_{w}~3.4E32`M`

particles of road dust on earth, and_{w}/M_{v}3E27~`d`

in a billion worlds: the ink drops.~3E36Pûrvavidehahas a måss of`9000`

yojanas, or`M`

atoms. Then there can be_{p}~9E31`M`

particles of road dust in such a land._{p}/M_{v}8E26~`d*1000`

such lands are reduced to dust this translates to`k`

kalpas at best.~2.35E66And so a challenge set to mathematicians two millennia ago has finally been answered. Buddhists may be glib talkers when it comes to numbers, but we do know the end and bounds of all their lands!

Furthermore, what strikes us in this passage from the Lotus Sutra is that a higher type of recursion – a repetition of numerations as we find it rising to a power tower in the Avatamsaka Sutra – seems to hide just around the corner, waiting to be discovered.