# Math Adventures/Wheat and the Chessboard

This clever story inspires students to explore exponential growth.

According to legend, the king became very fond of chess and wanted to reward the person who invented the game. The king offered the inventor his choice of rewards, and the clever inventor replied:

Your Majesty I ask you for just one thing. Take your chessboard and place on the first square one grain of wheat. On the first day I will take this grain home to feed my family. On the second day place on the second square 2 grains for me to take home. On the third day cover the third square with four grains for me to take. Each day double the number of grains you give me until you have placed wheat on every square of the chessboard. Then my reward will be complete.

The King replied:

This sounds like a small price to pay for your invention of such a fascinating game, I will see that your request is granted immediately.

^{[1]}

## Activities

Calculate how many grains of wheat are placed on each chessboard square. How many total gains of wheat have been placed at the end of each day? Complete this simple tabulation:

(The calculation quickly becomes laborious. Plan to pause as the numbers grow and consider the questions posed below. Expect that few if any students will complete this chart to day 64.)

Day (Square Number) | Grains placed that day | Total number of grains |
---|---|---|

1 | 1 | 1 |

2 | 2 | 3 |

3 | 4 | 7 |

... | ... | ... |

64 | ... | ... |

If one grain of wheat has a mass of 0.065 gram, then how many grains of wheat are in one kilogram? (Answer, 15,384 grains per kilogram. Note that the definition of a grain given above is approximate, and other acceptable values will give slightly different acceptable results.)

On what day will the total number of grains exceed one kilogram in mass? (Answer, on day 14 the total number of grains is 16,383. This is 1,064 grams of wheat or 1.064 kilograms.)

An average car has a mass somewhat less than 2,000 kilograms. On what day will 2,000 kilograms of wheat have been delivered? (Answer, on day 25, a total of 33,554,431 grains will have been delivered. This is 2,181 kilograms of wheat, more that the mass of an average car.) How much wheat was delivered on day 25? How much did that single day’s wheat weigh? (Answers,16,777,216 grains, or about 1,091 kg of wheat. Note that the delivery on each single day is more that the total amount delivered on all the previous days.)

The aircraft carrier, USS Gerald R. Ford is the largest in the world. It weighs approximately 100,000 tonnes, or 10^{8} Kilograms. On what day will the mass of the total number of grains delivered exceed the mass of the USS Gerald R. Ford? (Answer, on day 41 a total of 2,199,023,255,551 grains of wheat will have been delivered, weighing approximately 10^{8} kilograms.)

In 2017 worldwide wheat production was approximately 772 million metric tonnes. On what day does the mass of the total number of grains exceed this worldwide wheat production? (Answer, 772 million metric tonnes is 7.72 x 10^{11} kg. On day 54 more than 1 x 10^{12} kg of wheat will have been placed on the chess board.)

How many years of the world’s wheat production does the total amount of wheat placed on the chess board represent? (Answer, the total amount of wheat on the chess board represents approximately 1,553 times the world’s total wheat production.)

## Tabulation

The complete tabulation is shown here for reference. How much does the total amount of wheat weigh? (Answer, A total of 18,446,744,073,709,551,615 grains have been placed by the end of the 64^{th} day. This is about 10^{15} kilograms. The mass of the earth is about 6 x 10^{24} kilograms, and the mass of the largest aircraft carrier is about 100,000 tonnes or 10^{8} kg. Therefore, the mass of the wheat delivered, while less that the mass of the earth, is about the mass of 10 million aircraft carriers.)

The number 18,446,744,073,709,551,615 is read as:
eighteen quintillion, four hundred and forty-six quadrillion, seven hundred and forty-four trillion, seventy-three billion, seven hundred and nine million, five hundred and fifty-one thousand and six hundred and fifteen.^{[2]}

Note that after day 50 some rounding of the numbers may take place if a calculator or spreadsheet is used for the calculation. This is likely if the numbers exceeded the accuracy of the spreadsheet used to generate the table.

Day (Square Number) | Grains placed that day | Total number of grains | Kilograms |
---|---|---|---|

1 | 1 | 1 | 0.00 |

2 | 2 | 3 | 0.00 |

3 | 4 | 7 | 0.00 |

4 | 8 | 15 | 0.00 |

5 | 16 | 31 | 0.00 |

6 | 32 | 63 | 0.00 |

7 | 64 | 127 | 0.01 |

8 | 128 | 255 | 0.02 |

9 | 256 | 511 | 0.03 |

10 | 512 | 1,023 | 0.07 |

11 | 1,024 | 2,047 | 0.13 |

12 | 2,048 | 4,095 | 0.27 |

13 | 4,096 | 8,191 | 0.53 |

14 | 8,192 | 16,383 | 1.06 |

15 | 16,384 | 32,767 | 2.13 |

16 | 32,768 | 65,535 | 4.26 |

17 | 65,536 | 131,071 | 8.52 |

18 | 131,072 | 262,143 | 17.04 |

19 | 262,144 | 524,287 | 34.08 |

20 | 524,288 | 1,048,575 | 68.16 |

21 | 1,048,576 | 2,097,151 | 136 |

22 | 2,097,152 | 4,194,303 | 273 |

23 | 4,194,304 | 8,388,607 | 545 |

24 | 8,388,608 | 16,777,215 | 1,091 |

25 | 16,777,216 | 33,554,431 | 2,181 |

26 | 33,554,432 | 67,108,863 | 4,362 |

27 | 67,108,864 | 134,217,727 | 8,724 |

28 | 134,217,728 | 268,435,455 | 17,448 |

29 | 268,435,456 | 536,870,911 | 34,897 |

30 | 536,870,912 | 1,073,741,823 | 69,793 |

31 | 1,073,741,824 | 2,147,483,647 | 139,586 |

32 | 2,147,483,648 | 4,294,967,295 | 279,173 |

33 | 4,294,967,296 | 8,589,934,591 | 558,346 |

34 | 8,589,934,592 | 17,179,869,183 | 1,116,691 |

35 | 17,179,869,184 | 34,359,738,367 | 2,233,383 |

36 | 34,359,738,368 | 68,719,476,735 | 4,466,766 |

37 | 68,719,476,736 | 137,438,953,471 | 8,933,532 |

38 | 137,438,953,472 | 274,877,906,943 | 17,867,064 |

39 | 274,877,906,944 | 549,755,813,887 | 35,734,128 |

40 | 549,755,813,888 | 1,099,511,627,775 | 71,468,256 |

41 | 1,099,511,627,776 | 2,199,023,255,551 | 142,936,512 |

42 | 2,199,023,255,552 | 4,398,046,511,103 | 285,873,023 |

43 | 4,398,046,511,104 | 8,796,093,022,207 | 571,746,046 |

44 | 8,796,093,022,208 | 17,592,186,044,415 | 1,143,492,093 |

45 | 17,592,186,044,416 | 35,184,372,088,831 | 2,286,984,186 |

46 | 35,184,372,088,832 | 70,368,744,177,663 | 4,573,968,372 |

47 | 70,368,744,177,664 | 140,737,488,355,327 | 9,147,936,743 |

48 | 140,737,488,355,328 | 281,474,976,710,655 | 18,295,873,486 |

49 | 281,474,976,710,656 | 562,949,953,421,311 | 36,591,746,972 |

50 | 562,949,953,421,312 | 1,125,899,906,842,623 | 73,183,493,945 |

51 | 1,125,899,906,842,624 | 2,251,799,813,685,247 | 146,366,987,890 |

52 | 2,251,799,813,685,248 | 4,503,599,627,370,495 | 292,733,975,779 |

53 | 4,503,599,627,370,496 | 9,007,199,254,740,991 | 585,467,951,558 |

54 | 9,007,199,254,740,992 | 18,014,398,509,481,983 | 1,170,935,903,116 |

55 | 18,014,398,509,481,984 | 36,028,797,018,963,967 | 2,341,871,806,233 |

56 | 36,028,797,018,963,968 | 72,057,594,037,927,935 | 4,683,743,612,465 |

57 | 72,057,594,037,927,936 | 144,115,188,075,855,871 | 9,367,487,224,931 |

58 | 144,115,188,075,855,872 | 288,230,376,151,711,743 | 18,734,974,449,861 |

59 | 288,230,376,151,711,744 | 576,460,752,303,423,487 | 37,469,948,899,723 |

60 | 576,460,752,303,423,488 | 1,152,921,504,606,846,975 | 74,939,897,799,445 |

61 | 1,152,921,504,606,846,976 | 2,305,843,009,213,693,951 | 149,879,795,598,890 |

62 | 2,305,843,009,213,693,952 | 4,611,686,018,427,387,903 | 299,759,591,197,780 |

63 | 4,611,686,018,427,387,904 | 9,223,372,036,854,775,807 | 599,519,182,395,560 |

64 | 9,223,372,036,854,775,808 | 18,446,744,073,709,551,615 | 1,199,038,364,791,120 |

## Notes

- ↑ The Rice and Chessboard Legend, Mathscareers.org
- ↑ Tulengua.es