John Palermo, a newly-divorced Chicago sportswriter, tries to adjust to being a weekday bachelor with his office buddies, Robert Piccolo and David Keller, and weekend father with his two daughters, Ruth, 9 and Cathy, 15.

The Second Half - Netflix

Type: Scripted

Languages: English

Status: Ended

Runtime: 30 minutes

Premier: 1993-09-07

The Second Half - Wheat and chessboard problem - Netflix

The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as:

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?

The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 + ... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615, much higher than most expect. The problem appears in different stories about the invention of chess. One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by Ibn Khallikan. Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed. Macdonnell also investigates the earlier development of the theme. [According to al-Masudi's early history of India], shatranj, or chess was invented under an Indian king, who expressed his preference for this game over backgammon. [...] The Indians, he adds, also calculated an arithmetical progression with the squares of the chessboard. [...] The early fondness of the Indians for enormous calculations is well known to students of their mathematics, and is exemplified in the writings of the great astronomer Āryabaṭha (born 476 A.D.). [...] An additional argument for the Indian origin of this calculation is supplied by the Arabic name for the square of the chessboard, (بيت, “beit”), 'house'. [...] For this has doubtless a historical connection with its Indian designation koṣṭhāgāra, 'store-house', 'granary' [...]. This exercise can be used to demonstrate how quickly exponential sequences grow, as well as to introduce exponents, zero power, capital-sigma notation and geometric series. Updated for modern times using pennies and the hypothetical question, “Would you rather have a million dollars or the sum of a penny doubled every day for a month?”, the formula has been used to explain compounded interest.

The Second Half - Use - Netflix

Carl Sagan titled the second chapter of his final book The Persian Chessboard and wrote that when referring to bacteria, “Exponentials can't go on forever, because they will gobble up everything.” Similarly, The Limits to Growth uses the story to present suggested consequences of exponential growth: “Exponential growth never can go on very long in a finite space with finite resources.”

The Second Half - References - Netflix