Once upon a time, many centuries ago in India, a sage gifted a game he invented to a king. The game was चतुरंग (chaturang; Sanskrit for chess). चतुरंगी सेना (chaturangi sena) means an army consisting of all the four units: infantry, cavalry, chariots and elephants. 

 

As the legend goes, the king was extremely delighted to receive this game. He told the sage to ask for any gift. The sage asked the king to give him grains of wheat that would fill the chess board in a manner that the first square contains one grain, the second two grains, the third four grains, the fourth eight grains, and so on. He was doubling the number of grains in each successive square till he reached the last square i.e. the sixty-fourth. 

The king smiled, he thought the request was quite simple. The sage could have asked for anything expensive or precious, he wondered. But he soon realized that even the entire annual wheat production of Bharat Varsha (the then undivided India) would not satisfy the sage’s demand. In fact, it was higher than the entire world’s wheat production over 2,000 years. The king was worried that he would not be able to honour his word. But the kind sage forgave him. He went on to admit that he was more delighted with the sage’s mathematical tricks than the game he had originally gifted to him. 

This story has been very popular since many centuries. Even Nobel laureate physicist, George Gamow had written about it in his book "One, Two, Three...Infinity".

How much do you think is the total number of grains the sage demanded? 

It is 18446744073709551615. 

You must be wondering how I arrived at the correct answer. It is thanks to ancient Indian mathematics.

Here, the number of grains in each successive square is doubled. That is, the common ratio amongst the number of grains in successive squares is 2. Such sequences of numbers which bear a common ratio between successive terms are called geometric progression. 

Ancient Indian mathematicians had a fascination for large numbers and calculations. The first mathematician to give the formula for the sum of numbers in a geometric progression was a Jain mathematician Mahavir (815-877 AD) in his book Ganitasarasangraha written around 850 AD. Then, it was Bhaskar II (born 1114 AD) who gave the formula for the sum of terms in geometric progression in his book Lilavati, written around 1150 AD.

Let us now understand how they derived the formula:

Let 'a' be the first term in the series

'r' be the common ratio 

'n' be the number of terms in the series

and 'S' be the sum of the series.

We have,

S = a+ar+ar^2+ar^3+ar^4+⋯+ar^(n-1)        - (1)

rS = ar+ar^2+ar^3+ar^4+⋯+ar^(n-1)+ar^n        - (2)

Subtracting (1) from (2), we get

rS-S = ar^n-a

S(r-1) = a(r^n-1)

S = a(r^n-1)/(r-1)

In our chess problem, a = 1, r = 2 and n = 64. Hence, we get

S = 2^64-1 = 18446744073709551615

Bhaskaracharya illustrates the above formula with an interesting problem in Lilavati.

पूर्वं वराटकयुगं येन द्विगुणोत्तरं प्रतिज्ञातम् ।  

प्रत्यहमर्थिजनाय स मासे निष्कान् ददाति कति ।।

A gentleman gave 2 C (cowries) in charity the first day and thereafter the number of cowries he gave everyday was twice of what he gave the previous day. How many N (niskas) did he give away in one month?

Cowries (shells) and niskas were the prevalent currencies during the time of Bhaskaracharya. 

Listed below is the conversion table for the currencies

20 cowries = 1 kakini

40 kakinis = 1 pana

16 panas = 1 dramma

16 dramma = 1 niska

1280 cowries = 1 dramma

Solution:

Here, a = 2, r = 2, and n = 30.

S = 2(2^30-1)/(2-1) = 2(107374184-1) = 2147483646 C

= 104857 N, 9 drammas, 9 panas, 2 kakinis, 6 C

Well, what is the significance of this in today's times?

Containing the spread of Coronavirus is the top-most priority at this point in time. Epidemics grow exponentially. Let me illustrate this.

Suppose, a person is infected with COVID-19 and he in turn infects two of his friends. Each of these two friends in turn infect two of their friends. Each of these four people in turn infect two of their near and dear ones and so on. And, we have just seen how quickly this can become a huge number.

Hence, to contain the spread of Coronavirus or at least to slow it down, maintaining physical distancing is the best policy.

Let me end with a quote on ancient Indian Mathematics by a Swiss-American historian of mathematics, Florian Cajori (1859 - 1930)

"The Hindus solved problems in interest, discount, partnership, alligation*, summation of arithmetical and geometric series, and devised rules for determining the numbers of combinations and permutations. It may here be added that chess, the profoundest of all games, had its origin in India."

(*Alligation is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price or quality)

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• The History Good to Know 65
• Ancient Indian Mathematics65

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