>[!abstract] >If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish? ("Wheat and chessboard problem", 2025). The wheat-and-chessboard problem is an early illustration of how unintuitive geometric growth is to humans. ## Mathematical explanation The problem can be expressed as the geometric series $s$: $s = \sum_{n=0}^{63} 2^k = 2^0 + 2^1 + 2^2 + \cdots + 2^{63}$ Let's multiply each side by $2$, which increases each exponent $k$ by one: $2s = 2^1 + 2^2 + 2^3 + \cdots + 2^{64}$ Let's now subtract the original series $s$ from both sides: $2s - s = 2^1 + 2^2 + 2^3 + \cdots + 2^{64} - 2^0 - 2^1 - 2^2 - \cdots - 2^{63}$ The terms cancel out except for $2^{64}$ and $-2^0$: $s = 2^{64} - 2^0 = 2^{64} - 1$ Which gives over 18 quintillion ($18 \times 10^{18}$) grains of rice: $s = 18,446,744,073,709,551,615$ This is one order of magnitude larger than all the grains of sand on earth ($\approx 7.5 \times 10^{18}$). ## Origins There are at least two parables referring to this problem: ### The Northern India parable of Sissa >[!quote] >Our tendency to overlook exponential growth has been known for millennia. According to an Indian legend, the brahmin Sissa ibn Dahir was offered [by King Shihram around the 6th century CE] a prize for inventing an early version of chess. He asked for one grain of wheat to be placed on the first square on the board, two for the second square, four for the third square, doubling each time up to the 64th square. The king apparently laughed at the humility of ibn Dahir’s request – until his treasurers reported that it would outstrip all the food in the land (18,446,744,073,709,551,615 grains in total) (Robson, 2023). ### The Southern India parable of Krishna >[!quote] >Krishna once appeared in the form of a sage in the court of the king who ruled the region (Chembakassery) and challenged him for a game of chess (or chaturanga). The king being a chess enthusiast himself gladly accepted the invitation. The prize had to be decided before the game and the king asked the sage to choose his prize in case he won. The sage told the king that he had a very modest claim and being a man of few material needs, all he wished was a few grains of rice. The amount of rice itself would be determined using the chess-board in the following manner: one grain of rice on the first square, two grains in the second square, four in the third square, eight in the fourth square, sixteen in fifth square, doubling up to the final, sixty-fourth square. > >The king lost the game and the sage demanded the agreed-upon prize. As he started adding grains of rice to the chess board, the king soon realised the true scale of the sage's demands. The royal granary soon ran out of grains of rice. The king realised that he would never be able to fulfill the promised reward as the number of grains was increasing in a geometric progression and the total amount of rice required for a 64-square chess board was 18,446,744,073,709,551,615 grains, translating to trillions of tons of rice. > >Upon seeing the dilemma, the sage appeared to the king in his true-form and told the king that he did not have to pay the debt immediately but could pay him over time. The king would serve paal-payasam (pudding made of rice) in the temple freely to the pilgrims every day until the debt was paid off [Ambalappuzha Sree Krishna Swamy Temple in Kerala] ("Ambalappuzha Sree Krishna Swamy Temple", 2025). ## References - Ambalappuzha Sree Krishna Swamy Temple. (2025, April 06). In *Wikipedia*. https://en.wikipedia.org/w/index.php?title=Ambalappuzha_Sree_Krishna_Swamy_Temple&oldid=1280412096 - Robson, D. (2023, March 3). _Exponential growth bias: The numerical error behind covid-19_. BBC News. https://www.bbc.com/future/article/20200812-exponential-growth-bias-the-numerical-error-behind-covid-19 - Wheat and chessboard problem. (2025, April 06). In *Wikipedia*. https://en.wikipedia.org/w/index.php?title=Wheat_and_chessboard_problem&oldid=1282691382 ## Related - [[Exponential growth bias]] - [[Penny-doubling riddle]]