Math Adventures/Wheat and the Chessboard

Wheat and chessboard problem

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.

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.)

Grains each day

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 10⁸ 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 10⁸ 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 10¹¹ kg. On day 54 more than 1 x 10¹² 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 10¹⁵ kilograms. The mass of the earth is about 6 x 10²⁴ kilograms, and the mass of the largest aircraft carrier is about 100,000 tonnes or 10⁸ 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.

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.

Full Tabulation

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